Flexible framework for fluid topology optimization with OpenFOAM® and finite element-based high-level discrete adjoint method (FEniCS/dolfin-adjoint)

In order to implement the topology optimization method, it is necessary to simulate the fluid flow dynamics and also obtain the sensitivities with respect to the design variable (such as through the adjoint method). However, more complex fluid flows, such as turbulent, non-Newtonian, and compressible flows, may turn the implementation of these two aspects difficult and non-intuitive. In order to solve this deadlock, this work proposes the combination of two well-known and established open-source softwares: OpenFOAM (R) and FEniCS/dolfin-adjoint. OpenFOAM (R) already provides efficient implementations for various fluid flow models, while FEniCS, when combined with the dolfin-adjoint library, provides an efficient and automatic high-level discrete adjoint model. There have been various attempts for obtaining the adjoint model directly in OpenFOAM (R) , but they mostly rely on the following: (1) manually deducing the adjoint equations, which may become a hard and cumbersome task for complex models; (2) C++ automatic differentiation tools, which are generally computationally inefficient; and (3) finite differences, which have been developed for shape optimization (not topology optimization, where there are many more design variable values). Nonetheless, these approaches generally do not provide an easy setup, and may be fairly complex to consider. The FEniCS platform does not provide any fluid flow model out of the box, but makes it fairly simple to "simplistically" define them. The main problem of the FEniCS implementation and even implementations "by hand" (such as in C++, Matlab (R) or Python) is the convergence of the simulation, which would possibly require fairly complex adjustments in the implementation in order to reach convergence. Therefore, the combination proposed in this work (OpenFOAM (R) and FEniCS/dolfin-adjoint) is a simpler but efficient approach to consider more complex fluid flows, countering the difficult adjoint model implementation in OpenFOAM (R) and also the convergence issues in FEniCS. The implemented framework, referred as "FEniCS TopOpt Foam", can perform the coupling between the two softwares. Numerical examples are presented considering laminar and turbulent flows (Spalart-Allmaras model) for 2D, 2D axisymmetric, and 3D domains.