Integer programming topology optimization for subsonic compressible flows with geometry trimming

Abstract

The incompressibility hypothesis is successfully applied in a broad range of engineering problems simulated via computational fluid dynamics (CFD) codes. However, depending on the Mach number, the effects of compressibility are no longer negligible. Recently, the first papers about topology optimization considering compressible flows were published. All of them use the continuous design variables method which has some known drawbacks. The present work proposes topology optimization considering the compressible Navier-Stokes equations and discrete variables with a step of geometry trimming. This approach constrains the design variables to a binary set and solves the problem by using sequential integer linear programming, being a feasible alternative to classic density-based methods. Additionally, the solid regions are trimmed out from the physics analysis. A material model is inserted in the governing equations in such a way that, it is possible to calculate the sensitivity and, at the same time, guarantee the adiabatic condition. In this work, it is observed that the minimization of entropy generation as the objective function tends to converge to an undesired trivial solution. Alternatively, this work proposes the maximization of mass flow rate as the objective function to, indirectly, minimize the entropy. Furthermore, it is proposed the use of a new set of inlet boundary conditions. Three optimization cases are solved: the double-pipe, the pipe-bend, and an entropy maximization problem. The optimized solutions for the pipe-bend and double-pipe problems present geometries where the constraints are satisfied with the minimum entropy generation. Using the double-pipe example, it is conducted a study to understand the impacts of TOBS-GT parameters in the solutions. The results evidenced that, in some conditions, the implementation is prone to fall in a local minimum. In the third example, entropy maximization subjected to volume constraint is successfully solved. Moreover, a minimization case is also solved to aid in the interpretation of the results.