An R-adaptive Algorithm Based on Self-Organizing Maps for Solving Incompressible Flows with High-Order Discontinuous Galerkin Methods

Mesh quality is critical for the numerical accuracy of CFD (Computational Fluid Dynamics). Although various techniques have been developed to improve mesh applicability to complex flows, r-adaptive methods have received far too little attention. This paper introduces an r-adaptive algorithm based on the self-organizing maps (SOM) of Kohonen and applies it to unsteady CFD applications using the Discontinuous Galerkin (DG) method. Inspired by the properties of the DG method, the numerical discontinuity over the element interface is quantified as an adaptation indicator. The method requires only one adaptation, followed by solution projection onto the new mesh to save computational costs in unsteady cases. A significant advantage of this method is its higher efficiency without solving complex equations during the adaptation. The performance of the developed mesh adaptation algorithm is tested on the steady laminar viscous flow past a circular cylinder at Re=40 and the unsteady laminar viscous flow past a circular cylinder at Re=100 and two circular cylinders in a side-by-side arrangement at Re=200. Significant improvements in flow contours and coefficients are obtained after adaptation.