Combining High-Order Metric Interpolation and Geometry Implicitization for Curved r-Adaption

• Published in: Computer aided design Vol. 157; p. 103478
• Main Authors: Aparicio-Estrems, Guillermo, Gargallo-Peiró, Abel, Roca, Xevi
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-04-2023
• Subjects: [formula omitted]-adaption; Curved high-order meshes; Mesh optimization; Mesh optimization; r-adaption; Curved high-order meshes
• ISSN: 0010-4485; 1879-2685
• DOI: 10.1016/j.cad.2023.103478

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.
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•Second-order optimization for simultaneous curved r-adaption to metric and geometry.•Objective function accounting for metric and geometry deviations.•Implicit model representation for 2D and 3D NURBS models.•First- and second-order derivatives of the implicit representation of the model.•First- and second-order derivatives of a log-Euclidean high-order metric interpolation.