Airfoil Analysis and Optimization Using a Petrov–Galerkin Finite Element and Machine Learning

Abstract

For the analysis of low-speed incompressible fluid dynamics with turbulence around airfoils, we developed a finite element formulation based on a stabilized pressure and velocity formulation. To shape the optimization of bidimensional airfoils, this formulation is applied using machine learning (TensorFlow) and public domain global optimization algorithms. The goal is to maximize the lift-over-drag ratio by using the class-shape function transformation (CST) parameterization technique and machine learning. Specifically, we propose equal-order stabilized three-node triangles for the flow problem, standard three-node triangles for the approximate distance function (ADF) required in the turbulence stage, and stabilized three-node triangles for the Spalart–Allmaras turbulence model. The backward Euler time integration was employed. An implicit time-integration algorithm was adopted, and a solution was obtained using the Newton–Raphson method. This was made possible in the symbolic form via Mathematica with the AceGen package. Three benchmarks are presented, with Reynolds numbers up to 1×107, demonstrating remarkable robustness. After the assessment of the new finite element, we used machine learning and global optimization for four angles of attack to calculate airfoil designs that maximized CL/CD.