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Isogeometric analysis using manifold-based smooth basis functions

机译:使用基于流形的光滑基函数进行等几何分析

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We present an isogeometric analysis technique that builds on manifold-based smooth basis functions for geometric modelling and analysis. Manifold-based surface construction techniques are well known in geometric modelling and a number of variants exist. Common to most is the concept of constructing a smooth surface by blending together overlapping patches (or, charts), as in differential geometry description of manifolds. Each patch on the surface has a corresponding planar patch with a smooth one-to-one mapping onto the surface. In our implementation, manifold techniques are combined with conformal parameterisations and the partition-of-unity method for deriving smooth basis functions on unstructured quadrilateral meshes. Each vertex and its adjacent elements on the surface control mesh have a corresponding planar patch of elements. The star-shaped planar patch with congruent wedge-shaped elements is smoothly parameterised with copies of a conformally mapped unit square. The conformal maps can be easily inverted in order to compute the transition functions between the different planar patches that have an overlap on the surface. On the collection of star-shaped planar patches the partition of unity method is used for approximation. The smooth partition of unity, or blending functions, are assembled from tensor-product b-spline segments defined on a unit square. On each patch a polynomial with a prescribed degree is used as a local approximant. In order to obtain a mesh-based approximation scheme the coefficients of the local approximants are expressed in dependence of vertex coefficients. This yields a basis function for each vertex of the mesh which is smooth and non-zero over a vertex and its adjacent elements. Our numerical simulations indicate the optimal convergence of the resulting approximation scheme for Poisson problems and near optimal convergence for thin-plate and thin-shell problems discretised with structured and unstructured quadrilateral meshes. (C) 2016 Elsevier B.V. All rights reserved.
机译:我们提出了一种等几何分析技术,该技术基于基于流形的平滑基础函数进行几何建模和分析。基于歧管的表面构造技术在几何建模中是众所周知的,并且存在许多变体。最常见的是通过将重叠的面片(或图表)混合在一起来构造光滑表面的概念,如歧管的微分几何描述中所述。表面上的每个贴片都有一个对应的平面贴片,该贴片上有一个平滑的一对一映射。在我们的实现中,流形技术与共形参数化和整体划分方法相结合,用于在非结构化四边形网格上导出平滑基函数。曲面控制网格上的每个顶点及其相邻元素都具有相应的元素平面拼块。具有一致的楔形元素的星形平面补丁可通过共形映射的单位正方形的副本进行平滑参数化。共形图可以很容易地反转,以便计算在表面上有重叠的不同平面斑块之间的过渡函数。在收集星形平面斑块时,采用统一法进行分区。单位或混合函数的平滑划分是由在单位正方形上定义的张量积b样条线段组装而成的。在每个面片上,将具有规定次数的多项式用作局部近似值。为了获得基于网格的近似方案,根据顶点系数表达局部近似值的系数。这为网格的每个顶点产生了一个基函数,该基函数在一个顶点及其相邻元素上是平滑且非零的。我们的数值模拟表明,泊松问题所得近似方案的最优收敛性,以及用结构化和非结构化四边形网格离散的薄板和薄壳问题的最优收敛性。 (C)2016 Elsevier B.V.保留所有权利。

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