Mathematical Foundations of Isogeometric Analysis

Abstract

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 (FEM) and the boundary element method (BEM) in the frame of isogeometric analysis (IGA).

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications

“…We have called it Effective Grading (EG) strategy as the transition between coarser and finer regions is rather gradual and smooth in the LR meshes produced, with strict bounds on the aspect ratio of the boxes and on the sizes of the neighboring boxes. Such a grading ensures that the requirements on the mesh appearance listed in the axioms of adaptivity [5,4] are verified. The latter are a set of sufficient conditions on mesh grading, refinement strategy, error estimates and approximant spaces in an adaptive numerical method to theoretically guarantee optimal algebraic convergence rate of the numerical solution to the real solution.…”

“…Such a spanning completeness is more demanding to achieve in terms of meshing constraints and regularity, respectively, for (truncated) hierarchical B-splines and splines over T-meshes [18,12,3,8]. The grading properties are instead required to theoretically ensure optimal algebraic rates of convergence in adaptive IgA methods [5,4], even in presence of singularities in the PDE data or solution, similarly to what happens in Finite Element Methods (FEM) [19]. More specifically, the LR meshes generated by the proposed strategy satisfy the requirements listed in the axioms of adaptivity [5] in terms of grading and overall appearance.…”

Automatic Recognition of Financial Instruments Based on Anisotropic Partial Differential Equations

“…Tilt correction can restore the angle of scanned and photographed tax bill images, laying the foundation for subsequent image processing. The method first constructs the energy general function that meets the image requirements, then uses the variational theory to find the corresponding variational partial differential 3 Advances in Mathematical Physics equation, and finally introduces the time parameter for numerical solution to achieve the best processing of the target image [17]. The denoising method based on the variational theory usually needs to consider the properties of the target image, then seeks the most ideal energy general function form by studying these properties, represents the contents of the image by the spatial parametrization, and finally obtains the corresponding partial differential equation for solving to obtain better denoising performance.…”