## 11.10. Electromagnetic Tangential Vector Elements

In electromagnetics, we encounter serious problems when node-based
elements are used to represent vector electric or magnetic fields.
(See Electric Scalar Potential.) Node-based elements require
special treatment for enforcing boundary conditions of electromagnetic
field at material interfaces, conducting surfaces and geometric corners.
Tangentially continuous vector elements or edge elements, whose degrees
of freedom are associated with the edges of the finite element mesh,
have been shown to be free of such shortcomings. ([414])

### 11.10.1. Tetrahedral Elements

The tetrahedral element is the simplest tessellated shape and
is able to model arbitrary 3-D geometric structures. It is also well
suited for automatic mesh generation. The tetrahedral element, by
far, is the most popular element shape for 3-D applications in FEA.

For the 1st-order tetrahedral element (KEYOPT(1) = 1), the degrees
of freedom (DOF) are at the edges of element i.e., (DOFs = 6) (Figure 11.21: 1st-Order Tetrahedral Element). In terms of volume coordinates, the vector
basis functions are defined as:

where:

h_{IJ} = edge length between node I and
J |

λ_{I}, λ_{J}, λ_{K}, λ_{L} =
volume coordinates (λ_{K} = 1 - λ_{I} - λ_{J} - λ_{L}) |

λ_{I},
λ_{J},
λ_{K},
λ_{L} = the gradient of volume coordinates |

The tangential component of the approximated field is constant
along the edge. The normal component of field varies linearly.

### 11.10.2. Hexahedral Elements

Tangential vector bases for hexahedral elements can be derived
by carrying out the transformation mapping a hexahedral element in
the global xyz coordinate to a brick element in local str coordinates.

For the 1st-order brick element (KEYOPT(1) = 1), the degrees
of freedom (DOF) are at the edges of element (DOFs = 12) (Figure 11.22: 1st-Order Brick Element). The vector basis functions are cast in the
local coordinate

where:

h_{s}, h_{t}, h_{r} = length of element edge |

s,
t,
r = gradient of local
coordinates |