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Generating high-quality high-order parameterization for isogeometric analysis on triangulations

机译：生成高质量的高阶参数化，以进行三角剖分的等几何分析

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摘要

This paper presents an approach for automatically generating high-quality high-order parameterizations for isogeometric analysis on triangulations. A B-spline represented boundary geometry is parameterized into a collection of high-order Bezier triangles or tetrahedra in 2D and 3D spaces, respectively. Triangular Bezier splines are used to represent both the geometry and physical fields over the triangulation. By imposing continuity constraints on the Bezier ordinates of the elements, a set of global C-r smooth basis is constructed and used as the basis for analysis. To ensure high quality of the parameterization, both the parametric and physical meshes are optimized to reduce the shape distortion of the high-order elements relative to well-defined reference elements. The shape distortion is defined based on the Jacobian of the triangular Bezier splines, and its sensitivity with respect to the location of control points is derived analytically and evaluated efficiently. Moreover, a sufficient condition is derived to guarantee the generated mesh is free of local self-intersection, thanks to the convex hull property of triangular Bezier splines. By using a Heaviside projection function, the non-negative Jacobian determinant constraints are formulated efficiently as a single optimization constraint. Several 2D and 3D numerical examples are presented to demonstrate that high-quality high-order elements are generated using our approach. (C) 2018 Elsevier B.V. All rights reserved.
机译：本文提出了一种自动生成高质量高阶参数化的方法，用于三角剖分的等几何分析。将B样条表示的边界几何参数化为分别在2D和3D空间中的高阶Bezier三角形或四面体的集合。三角贝塞尔曲线样条线用于表示三角剖分上的几何场和物理场。通过在元素的贝塞尔坐标上施加连续性约束，可以构建一组全局C-r平滑基础并将其用作分析的基础。为了确保高质量的参数化，对参数网格和物理网格都进行了优化，以减少高阶元素相对于定义明确的参考元素的形状失真。基于三角贝塞尔曲线样条的雅可比行列式定义形状失真，并通过分析得出其对控制点位置的敏感性并进行有效评估。此外，由于三角Bezier样条的凸包属性，导出了充分的条件以确保生成的网格没有局部自相交。通过使用Heaviside投影函数，可将非负Jacobian行列式约束有效地公式化为单个优化约束。给出了几个2D和3D数值示例，以证明使用我们的方法可以生成高质量的高阶元素。 （C）2018 Elsevier B.V.保留所有权利。

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