690 research outputs found

    Dimensions of Biquadratic and Bicubic Spline Spaces over Hierarchical T-meshes

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    This paper discusses the dimensions of biquadratic C1 spline spaces and bicubic C2 spline spaces over hierarchical T-meshes using the smoothing cofactor-conformality method. We obtain the dimension formula of biquadratic C1 spline spaces over hierarchical T-meshes in a concise way. In addition, we provide a dimension formula for bicubic C2 spline spaces over hierarchical T-mesh with fewer restrictions than that in the previous literature. A dimension formula for bicubic C2 spline spaces over a new type hierarchical T-mesh is also provided.Comment: 21 pages, 19 figure

    Inf-sup stability of isogeometric Taylor-Hood and Sub-Grid methods for the Stokes problem with hierarchical splines

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    In this article, we prove the inf-sup stability of an adaptive isogeometric discretization of the Stokes problem. The discretization is based on the hierarchical generalization of the isogeometric Taylor-Hood and Sub-Grid elements, which were described by Bressan & Sangalli (2013, Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique. IMA J. Numer. Anal., 33, 629- 651) for tensor-product splines. In order to extend the existing proof to the hierarchical setting, we need to adapt some of the steps considerably. In particular, the required local approximation estimate is obtained by analysing the properties of the quasi-interpolant of Speleers & Manni (2016, Effortless quasi-interpolation in hierarchical spaces. Numer. Math., 132, 155-184) with respect to certain Sobolev norms. In addition to the theoretical results, we also perform numerical tests in order to analyse the dependency of the inf-sup constant on the mesh regularity assumptions. Finally, the article also presents a numerical convergence test of the resulting adaptive method on a T-shaped domain

    An isogeometric analysis for elliptic homogenization problems

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    A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard Finite Element Heterogeneous Multiscale Method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rational B-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Beside the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. Numerical results show the excellent performance of the proposed method
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