Abstract
This paper presents an adjoint Lattice Boltzmann Method (LBM) coupled with the Level-Set Method (LSM) for topology optimization of thermal fluid flows. The adjoint-state formulation implies discrete velocity directions in order to take into account the LBM boundary conditions. These boundary conditions are introduced at the beginning of the adjoint-state method as the LBM residuals, so that the adjoint-state boundary conditions can appear directly during the adjoint-state equation formulation. The proposed method is tested with 3 numerical examples concerning thermal fluid flows, but with different objectives: minimization of the mean temperature in the domain, maximization of the heat evacuated by the fluid, and maximization of the heat exchange with heated solid parts. This latter example, treated in several articles, is used to validate our method. In these optimization problems, a limitation of the maximal pressure drop and of the porosity (number of fluid elements) is also applied. The obtained results demonstrate that the method is robust and effective for solving topology optimization of thermal fluid flows.
Highlights
The determination of optimal designs is an important objective in many engineering problems
A gradient-type method has been developed for topology optimization of thermal fluid flow problems
The forward problem is solved by Lattice Boltzmann Method (LBM), and the calculation of the augmented cost function gradient is performed with an adjoint-state method in order to deal with a large number of design variables
Summary
The determination of optimal designs is an important objective in many engineering problems. Yaji formulated a LBM fluid flow topology optimization method [37] with velocity discrete Boltzmann equation to derive the adjoint states, and Liu [38] used discrete adjoint-state LBM, but with the Multiple Relaxation Time (MRT) collision operator In both works, the LBM boundary conditions were very simple (bounce-back or periodic), while commonly-used velocity and pressure boundary conditions [19] were not treated. The major novelty of this paper lies in: (1) the solution of thermal fluid flow topology optimization problems using an adjoint-state LBM coupled with the LSM, (2) the new introduction of the LBM boundary conditions in the adjoint-state formulation, and (3) the use of the LBM incompressible model that improves the forward problem accuracy, and simplifies the adjoint-state calculations. In LBM, this equilibrium function is obtained by a second-order Taylor expansion of the maxwellian distribution
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