Defining metric-aware size-shape measures to validate and optimize curved high-order meshes
<|reference_start|>Defining metric-aware size-shape measures to validate and optimize curved high-order meshes: We define a regularized size-shape distortion (quality) measure for curved high-order elements on a Riemannian space. To this end, we measure the deviation of a given element, straight-sided or curved, from the stretching, alignment, and sizing determined by a target metric. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements on Riemannian spaces determined by constant and point-wise varying metrics. The examples illustrate that the distortion can be minimized to curve (deform) the elements of a given high-order (linear) mesh and try to match with curved (linear) elements the point-wise stretching, alignment, and sizing of a discrete target metric tensor. In addition, the resulting meshes simultaneously match the curved features of the target metric and boundary. Finally, to verify if the minimization of the metric-aware size-shape distortion leads to meshes approximating the target metric, we compute the Riemannian measures for the element edges, faces, and cells. The results show that, when compared to anisotropic straight-sided meshes, the Riemannian measures of the curved high-order mesh entities are closer to unit. Furthermore, the optimized meshes illustrate the potential of curved $r$-adaptation to improve the accuracy of a function representation.<|reference_end|>
@article{aparicio-estrems2024defining,
    title={Defining metric-aware size-shape measures to validate and optimize
  curved high-order meshes},
    author={Guillermo Aparicio-Estrems, Abel Gargallo-Peir'o, Xevi Roca},
    journal={Computer-Aided Design, Volume 168, March 2024, 103667},
    year={2024},
    doi={10.1016/j.cad.2023.103667},
    archivePrefix={arXiv},
    eprint={2403.13528},
    primaryClass={cs.CE}
}