Helicity functionals and metric invariance in three dimensions

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