Discontinuous Galerkin Based Isogeometric Analysis for Geometric flows
Discontinuous Galerkin Based Isogeometric Analysis for Geometric flows
复制标题
基于不连续伽辽金的几何流等几何分析
DOI：
10.1007/s10915-016-0307-5
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发表时间：
2017
影响因子：
2.5
通讯作者：
Chen F
中科院分区：
文献类型：
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作者：
Zhang F;Xu Yan;Chen F
提出了一种将二、四阶几何流的等几何分析与不连续伽辽金（DG）法相结合的方法来生成由多个斑块组成的整流面。该技术可用于解决几何建模中的一个具有挑战性的问题-将多个补丁平滑地粘合在一起以创建复杂的模型。非均匀有理b样条（NURBS）是计算机辅助设计中最流行的几何模型表示形式，用于描述几何形状和表示数值解。由于两个不同斑块上的NURBS基函数是独立的，因此可以适当地使用DG方法将多个斑块粘合在一起以获得光滑解。我们提出了半离散DG格式来解决该问题，并证明了该格式的稳定性。我们的方法有以下优点。首先，NURBS基函数的几何灵活性，特别是对多个补丁的使用，使我们能够构建具有复杂几何和拓扑结构的曲面模型。其次，构造几何是公平的。第三，由于只有NURBS补丁的控制点按照几何流进行演化，并且它们的数量（自由度）非常小，因此我们的算法非常高效。最后，该方法易于制定和实现。我们将该方法应用于平均曲率流和准表面扩散流中，以解决各种几何建模问题，如最小表面生成、表面混合和孔填充等。通过实例说明了该方法的有效性。
We propose a method which combines isogeometric analysis with the discontinuous Galerkin (DG) method for second and fourth order geometric flows to generate fairing surfaces, which are composed of multiple patches. This technique can be used to tackle a challenging problem in geometric modeling–gluing multi-patches together smoothly to create complex models. Non-uniform rational B-splines (NURBS), the most popular representations of geometric models developed in Computer Aided Design, are employed to describe the geometry and represent the numerical solution. Since NURBS basis functions over two different patches are independent, DG methods can be appropriately applied to glue the multiple patches together to obtain smooth solutions. We present semi-discrete DG schemes to solve the problem, and-stability is proved for the proposed schemes. Our method enjoys the following advantages. Firstly, the geometric flexibility of NURBS basis functions, especially the use of multiple patches, enable us to construct surface models with complex geometry and topology. Secondly, the constructed geometry is fair. Thirdly, since only the control points of the NURBS patches evolve in accordance with the geometric flows, and their number (degrees of freedom) is very small, our algorithm is very efficient. Finally, this method can be easily formulated and implemented. We apply the method in mean curvature flows and in quasi surface diffusion flows to solve various geometric modeling problems, such as minimal surface generation, surface blending and hole filling, etc. Examples are provided to illustrate the effectiveness of our method.