Mathematical Foundations of Adaptive Isogeometric Analysis

Abstract

This paper reviews the state of the art and discusses recent developments in the field of adaptive isogeometric analysis, with special focus on the mathematical theory. This includes an overview of available spline technologies for the local resolution of possible singularities as well as the state-of-the-art formulation of convergence and quasi-optimality of adaptive algorithms for both the finite element method and the boundary element method in the frame of isogeometric analysis.

Isogeometric analysis of the lithosphere under topographic loading: Igalith v1.0.0

“…In our work, we employ hierarchical B-splines as described in Vuong et al (2011). Adaptive local refinement can then be performed if an error estimator for the numerical solution to the problem is available (Garau and Vázquez, 2018;Buffa et al, 2022).…”

“…One advantage of using them for shell and plate problems is the simple construction of C 1 isogeometric spline spaces on a single patch to discretize the equations with less degrees of freedom than standard C 1 finite element methods. Further features and capabilities of isogeometric analysis presented in this paper are the exact representation of curved domains, the coupling of multiple patches, which preserves the C 1 continuity along the interfaces (Kapl et al, 2018;Farahat et al, 2023), and adaptive local refinement using hierarchical B-splines (Vuong et al, 2011;Garau and Vázquez, 2018;Buffa et al, 2022).…”

A short perspective on a posteriori error control and adaptive discretizations

“…It has been a very active research field, starting with the works of Dörfler (1996); Morin et al (2000). The combination with nonlinear approximation theory has been achieved in Binev et al (2004); Stevenson (2007), which lead to many further results for different methods/equations/algorithms, see for example Cascon et al (2008); Becker and Mao (2008); Bonito and Nochetto (2010); Ferraz-Leite et al ( 2010); Kreuzer and Georgoulis (2018); Buffa et al (2022). Now it has attained a state of maturity in the context of second-order linear elliptic partial differential equations, certain finite element methods and mesh refinement algorithms.…”

Isogeometric analysis with C1-smooth functions over multi-patch surfaces

“…As for the planar case, we base our numerical simulations on the hierarchical spline space consisting of truncated hierarchical splines (THB-splines) [78,79,80,77], and we use for all the presented examples an admissible mesh, c.f. [81], of class µ, with µ = 3.…”