Métodos de elementos finitos com adaptatividade de malha para escoamentos compressíveis e incompressíveis

Abstract

Numerical simulation of flows using the Finite Element Method faces severe challenges arising from the non-linearity of the governing equations, the predominance of convection, and the formation of strong local gradients, such as vortical structures, boundary layers, and shock discontinuities, inherent to the incompressible, compressible, and inviscid regimes of the Navier-Stokes and Euler equations. Although Adaptive Mesh Refinement (AMR) is essential to focus spatial resolution where necessary, conventional approaches depend on empirical parameter calibration by the user and suffer from the risk of uncontrolled growth in the number of elements. To overcome these limitations, this work proposes Auto AMR, an autonomous adaptivity methodology that integrates a posteriori error estimator with a statistical distribution function and introduces an original penalty function based on the ratio between active cells and the maximum mesh capacity, establishing a self-adjusting criterion that balances the reduction of numerical error and computational cost without manual intervention. This methodology is coupled with a variational formulation stabilized by techniques such as PSPG, SUPG, and Y Zβ shock capture, in addition to an implicit second-order BDF scheme for temporal integration. The approach is widely validated for incompressible and compressible Navier-Stokes models and for Euler equations through convergence studies and a range of classical problems, including square cavity, cylinder, step, doubly periodic shear layer, NACA 0012 airfoil, 2D isentropic vortex, oblique and reflected shocks, and diamond airfoil flows in supersonic and high Reynolds number regimes. The results demonstrate that Auto AMR autonomously tracks complex physical structures in multiple flow regimes, ensuring high accuracy with a considerable reduction in the number of elements compared to fixed meshes.