In the past decade frame fields have emerged as a promising approach for generating hexahedral meshes for CFD and CAE applications. One important problem asks for construction of a boundary-aligned frame field with prescribed singularity constraints over a volume that corresponds to a valid hexahedral mesh. We give a necessary and sufficient condition in terms of solutions to a system of monomial equations with variables in the binary octahedral group when a boundary frame field and singularity graph have been fixed. This is phrased with respect to general $CW$-decompositions of the volume, which allows some flexibility in simplifying these systems. Along the way we look at frame field design from an algebraic topological perspective, proving various results, some known, some new.

中文翻译：

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六角网格划分的框架场设计拓扑

在过去的十年中，框架场已成为为 CFD 和 CAE 应用生成六面体网格的一种很有前途的方法。一个重要的问题是在对应于有效六面体网格的体积上构建具有指定奇异性约束的边界对齐框架场。当边界框场和奇点图已经固定时，我们给出了二元八面体群中变量的单项式方程组的解的充分必要条件。这是相对于体积的一般 $CW$ 分解而言的，这允许在简化这些系统时具有一定的灵活性。在此过程中，我们从代数拓扑的角度看待框架场设计，证明了各种结果，有些是已知的，有些是新的。