This paper proposes a method to compute crossfields based on the Ginzburg-Landau theory. The Ginzburg-Landau functional has two terms: the Dirichlet energy of the distribution and a term penalizing the mismatch between the fixed and actual norm of the distribution. Directional fields on surfaces are known to have a number of critical points, which are properly identified with the Ginzburg-Landau approach: the asymptotic behavior of Ginzburg-Landau problem provides well-distributed critical points over the 2-manifold, whose indices are as low as possible. The central idea in this paper is to exploit this theoretical background for crossfield computation on arbitrary surfaces. Such crossfields are instrumental in the generation of meshes with quadrangular elements. The relation between the topological properties of quadrangular meshes and crossfields are hence first recalled. It is then shown that a crossfield on a surface can be represented by a complex function of unit norm with a number of critical points, i.e., a nearly everywhere smooth function taking its values in the unit circle of the complex plane. As maximal smoothness of the crossfield is equivalent with minimal energy, the crossfield problem is equivalent to an optimization problem based on Ginzburg-Landau functional. A discretization scheme with Crouzeix-Raviart elements is applied and the correctness of the resulting finite element formulation is validated on the unit disk by comparison with an analytical solution. The method is also applied to the 2-sphere where, surprisingly but rightly, the computed critical points are not located at the vertices of a cube, but at those of an anticube.

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计算交叉场——一种基于Ginzburg-Landau理论的偏微分方程方法

本文提出了一种基于Ginzburg-Landau 理论的交叉场计算方法。Ginzburg-Landau 泛函有两个项：分布的狄利克雷能量和惩罚分布的固定范数和实际范数之间的不匹配的项。已知表面上的方向场具有许多临界点，这些临界点可以通过 Ginzburg-Landau 方法正确识别：Ginzburg-Landau 问题的渐近行为在 2-流形上提供了分布良好的临界点，其指数同样低尽可能。本文的中心思想是利用这一理论背景在任意表面上进行交叉场计算。这种交叉场有助于生成具有四边形元素的网格。因此首先回顾四边形网格的拓扑特性和交叉场之间的关系。然后表明，表面上的交叉场可以由具有多个临界点的单位范数的复函数表示，即几乎处处取其值在复平面的单位圆中的平滑函数。由于交叉场的最大平滑度等效于最小能量，因此交叉场问题等效于基于 Ginzburg-Landau 泛函的优化问题。应用 Crouzeix-Raviart 单元的离散化方案，并通过与解析解进行比较，在单位圆盘上验证所得有限元公式的正确性。该方法也适用于 2 球体，其中令人惊讶但正确的是，