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首页> 外文期刊>IEEE Transactions on Magnetics >Helicity functionals and metric invariance in three dimensions
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Helicity functionals and metric invariance in three dimensions

机译：三维螺旋函数和度量不变性

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摘要

The solvability, gauge invariance, and topological aspects of dual variational principles shed light on the difficulties that arise from the use of a vector potential in three dimensions in place of a stream function in two dimensions. These aspects reveal the central role of a relative de Rham complex in place of the usual Tonti diagram. By considering the 'spin complex' associated with the de Rham complex, it is seen that the helicity functional enables the scalar potential to be used in the dual role of a Lagrange multiplier which fixes the gauge of the vector potential. The metric and constitutive law independence of the helicity term is considered. The main purpose is to show how the invariant terms of the helicity functional can be used to avoid rebuilding (reassembling) large parts of the finite-element stiffness matrix in iterative computations involving constitutive laws which change with every iteration. The results are phrased in terms of differential forms.
机译：对偶变分原理的可解性，规范不变性和拓扑方面阐明了因使用三维矢量势代替二维流函数而引起的困难。这些方面揭示了相对de Rham复合体取代通常的Tonti图的核心作用。通过考虑与de Rham复合体关联的“自旋复合体”，可以看到螺旋函数使标量势能用于固定向量势距的Lagrange乘法器的双重作用。考虑了螺旋度术语的公制和本构法独立性。主要目的是说明在涉及本构定律的迭代计算中，螺旋函数的不变项如何用于避免重建（重新组装）有限元刚度矩阵的大部分，该定律随每次迭代而变化。结果用微分形式表述。

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