hr-Adaptivity for nonconforming high-order meshes with the target matrix optimization paradigm

• Published in: Engineering with computers 2022-08, Vol.38 (4), p.3721-3737
• Main Authors: Dobrev, Veselin, Knupp, Patrick, Kolev, Tzanio, Mittal, Ketan, Tomov, Vladimir
• Format: Article
• Language: English
• Subjects: Aspect ratio; CAE) and Design; Calculus of Variations and Optimal Control; Optimization; Classical Mechanics; Computer Science; Computer-Aided Engineering (CAD; Control; Error reduction; Exact solutions; Grid refinement (mathematics); Math. Applications in Chemistry; Mathematical and Computational Engineering; Optimization; Original Article; Parameters; Quadrilaterals; Systems Theory; Topology
• ISSN: 0177-0667; 1435-5663
• DOI: 10.1007/s00366-021-01407-6

We present an hr -adaptivity framework for optimization of high-order meshes. This work extends the r -adaptivity method by Dobrev et al. (Comput Fluids, 2020), 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 rotation . 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 mesh nodes. 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.