The algorithm that is the basis of most of our adaptive work is MOESS: Mesh Optimization via Error Sampling and Synthesis.  Developed by Masayuki Yano[1][2] during his doctoral work in our group, MOESS solves an optimization problem to find a mesh producing the lowest error for a given computational cost.  Mesh-metric duality is a key concept used in MOESS to represent desired mesh properties in a continuous manner.  Our use of metrics to describe an unstructured mesh is based upon the work of Loseille & Alauzet at INRIA. This allows casting the discrete optimization of a mesh into a continuous optimization problem for a metric.  This optimal metric is then provided to a metric-based mesher which generates a mesh matching the metric.

In order to solve this continuous optimization, a model for the dependence of the error on the metric is required.  MOESS assumes that an error indicator exists which can be localized to a region in the domain.  For output error estimation, we localize the Dual-Weighted Residual (Rannacher et al) calculated with the solution  and the adjoint (using polynomial order p and p+1, respectively) on the current mesh.  An example for a transonic calculation over the RAE2822 airfoil is shown below.

The Error Sampling and Synthesis step involves locally subdividing the mesh, solving local problems on the enriched space, and synthesizing an error-metric model.  This process is shown below for the elemental sampling process we have used for discontinuous finite element discretization.  

 

MOESS has been extended to continuous finite element discretizations[3][4].

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References