Minimum constraints for finite element vector potential problems with Neumann boundary conditions

Finite element vector potential magnetostatic problems that are determined only by inhomogeneous Neumann boundary conditions, i.e., by specified tangent components of H on the boundary, are discussed. The minimum constraint condition required to render such matrix problems nonsingular is derived from the spurious mode properties of individual finite elements. It is shown that constraining three components of A at a single point does not remove all matrix singularities. When five additional constraints are applied, the remaining singular shear modes are removed, and the problem is nonsingular. Constraint techniques are demonstrated with an example.