Higher order 3D isoparametric finite elements for improved magnetic field calculation accuracy

The formulation and application of second-order isoparametric three-dimensional magnetic vector potential finite elements is described. For a hexahedral finite element, detailed expressions are derived for all the shape functions and their derivatives. A patch test is passed by hexahedrons of both first and second orders. An extremely coarse mesh of both element orders is used to model the pathological problem of a closed steel loop deformed in three-dimensional space. The first-order elements are shown to give only 2.4% of the correct magnetic field in the steel, while the second-order elements are shown to give 80 to 96% of the correct magnetic field.