  We detail how to use Newton's method for distortion-based curved $r$-adaption
to a discrete high-order metric field while matching a target geometry.
Specifically, we combine two terms: a distortion measuring the deviation from
the target metric; and a penalty term measuring the deviation from the target
boundary. For this combination, we consider four ingredients. First, to
represent the metric field, we detail a log-Euclidean high-order metric
interpolation on a curved (straight-edged) mesh. Second, for this metric
interpolation, we detail the first and second derivatives in physical
coordinates. Third, to represent the domain boundaries, we propose an implicit
representation for 2D and 3D NURBS models. Fourth, for this implicit
representation, we obtain the first and second derivatives. The derivatives of
the metric interpolation and the implicit representation allow minimizing the
objective function with Newton's method. For this second-order minimization,
the resulting meshes simultaneously match the curved features of the target
metric and boundary. Matching the metric and the geometry using second-order
optimization is an unprecedented capability in curved (straight-edged)
$r$-adaption. This capability will be critical in global and cavity-based
curved (straight-edged) high-order mesh adaption.

Computational Engineering, Finance, and Science