  We present an $hr$-adaptivity framework for optimization of high-order
meshes. This work extends the $r$-adaptivity method for mesh optimization by
Dobrev et al., where we utilized the Target-Matrix Optimization Paradigm (TMOP)
to minimize a functional that depends on each element's current and target
geometric parameters: element aspect-ratio, size, skew, and orientation. Since
fixed mesh topology limits the ability to achieve the target size and
aspect-ratio at each position, in this paper we augment the $r$-adaptivity
framework with nonconforming adaptive mesh refinement to further reduce the
error with respect to the target geometric parameters. The proposed
formulation, referred to as $hr$-adaptivity, introduces TMOP-based quality
estimators to satisfy the aspect-ratio-target via anisotropic refinements and
size-target via isotropic refinements in each element of the mesh. The
methodology presented is purely algebraic, extends to both simplices and
hexahedra/quadrilaterals of any order, and supports nonconforming isotropic and
anisotropic refinements in 2D and 3D. Using a problem with a known exact
solution, we demonstrate the effectiveness of $hr$-adaptivity over both $r-$
and $h$-adaptivity in obtaining similar accuracy in the solution with
significantly fewer degrees of freedom. We also present several examples that
show that $hr$-adaptivity can help satisfy geometric targets even when
$r$-adaptivity fails to do so, due to the topology of the initial mesh.
