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.

翻译：在过去十年中,框架字段已成为为CFD和CAE应用生成六相色片的一种很有希望的方法。一个重要问题是要求建立一个边界对齐框架字段,对相当于有效六相网状体积的体积规定单一性限制。当确定了一个边界框架字段和单相图时,我们给一个具有二进制八相形组变量的单一方程式系统提供了必要和充分的解决方案条件。这是针对该体积一般的 $CW$的分解而设计的,允许在简化这些系统方面有一定的灵活性。与此同时,我们从代数表层学的角度看待框架场设计的方式,证明了各种结果,有些是已知的,有些是新的。