Topology optimization of fluid flow by using Integer Linear Programming

Topology optimization of fluid flow problems is still a challenging open problem, especially when considering turbulence, compressibility, or the addition of different physics. In the current implementation of topology optimization for fluids considering density methods, there are essentially three problems. First, the grayscale in the result makes it difficult to identify the precise contour of the fluid region, which may be a problem in some applications and during the optimization process as well. Second, even for low Reynolds flow design problems, a continuation scheme of the material model penalization parameters is necessary to avoid a grayscale and to obtain a clear boundary. Third, in complex fluid flow optimization problems, it is difficult to specify the maximum value of the inverse permeability to avoid the fluid to flow inside the solid. This work proposes a novel methodology that tackles the first two problems, i.e., it avoids the grayscale and obtains clear boundaries. The goal of this work is to implement the Topology Optimization of Binary Structures (TOBS) (Sivapuram and Picelli, Finite Elem Anal Des 139:49–61, 2018) for fluid flow design, which is a novel topology optimization method that has been used in solid mechanics to generate optimized structural solutions considering only binary {0,1} design variables. The main advantage of {0,1} methods is the clear definition of the interface and the absence of grayscale. It is a method easy to implement which preserves the material distribution features. Some classic fluid problems are considered to illustrate the problem, such as the double channel and the bend pipe, and also a more complex example that usually presents grayscale issues, which is the fluid diode design. The optimization results show the feasibility of the TOBS when applied to fluid flow problems. The physical problem is solved by using the finite element method and the optimization problem with CPLEXⒸ, a proprietary optimization package from IBM. The present work successfully eliminates the grayscale problem, bringing clear boundaries in the interface fluid-solid.