Magic formulas are the geometric identities at the root of modern compatible schemes for polyhedral grids. We present rigorous yet elementary proofs of the magic formulas originating from Stokes theorem. The proofs enlighten new fundamental aspects of the mass matrices produced with the magic formulas. First, the construction of the mass matrices works for an unexpectedly broad type of mesh cells. Second, they show that dual nodes can be arbitrarily positioned thus extending the construction of the dual barycentric grid.

中文翻译：

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新的魔术公式演示显示了多面体网格的几何定义矩阵的意外特征

魔术公式是现代多面体网格兼容方案的根本几何标识。我们提供源自斯托克斯定理的魔术公式的严格而基本的证明。证明启发了使用魔术公式生成的质量矩阵的新基本方面。首先，质量矩阵的构造适用于出乎意料的广泛类型的网格单元。其次，它们表明可以任意放置双节点，从而扩展了双重心网格的构造。