An adaptive mesh refinement approach based on optimal sparse sensing

Abstract

We introduce a new approach for adaptive mesh refinement in which adaptivity is driven by low rank decomposition and optimal sensing of the dynamically evolving flow field. This method seeks an ordered set of locations for mesh adaptation from the instantaneous data-driven basis of an online proper orthogonal decomposition of the velocity, which organizes features into sparse optimal orthogonal modes based on an energy norm. The sensing is achieved via a computationally expedient discrete empirical interpolation method using rank-revealing QR factorization (Drmac and Gugercin SIAM J Sci Comput 38(2):A631–A648, 2016). The methodology is applicable to a wide range of numerical discretizations, and is tested on a spatiotemporally evolving incompressible turbulent jet, a complex wind turbine wake, and supersonic flow over a forward-facing step. The proposed approach is demonstrated to predict accurate velocity statistics and yield significantly smaller grids in comparison to gradient-based methods. The algorithm is seen to focus refinement in the vicinity of dynamically significant regions such as those characterized by high turbulence kinetic energy, coherent structures and shock interactions. Moreover, the approach does not require parameters or thresholds, which may be difficult to obtain for complex flows, to be known a priori to facilitate mesh adaptation.