FIGURE 44.1 Three-dimensional second-order Isoparametric finite elements. (a) Left, master tetrahedron, 10 nodes in al space(E, n, 4); right, actual tetrahedron, 10 nodes in global space (x, y, z).(b)Left, master hexahedron, 20 nodes in local space(S, n, S); right, actual hexahedron, 10 nodes in global space(x, y, z) Although the FEM is widely used by industry for electromagnetic problems covering the entire frequency range, there are many situations where special methods are more effective. This is particularly the case for high frequency problems, e.g., millimeter and microwave integrated circuit structures where integral equation techniques and such procedures as transmission line modeling(TLM), spectral domain approach, method of lines, and wire grid methods are often preferred [Itoh, 1989](see Chapter 43) Edge elements Using potentials and nodal finite elements(see Fig. 44. 1)rather than field components directly has the advantage that difficulties arising from field discontinuities at material interfaces can be avoided. However, if the element basis functions [see Eq (44.5)] are expressed in terms of the field(H, say) constrained along an element edge, sen tangential field continuity is enforced [Bossavit, 1988]. The field equations [Eq.(44.1)] in terms of the ield intensity for the low frequency limit reduce to V×V×H+aH 0 (44.7) t c 2000 by CRC Press LLC© 2000 by CRC Press LLC Although the FEM is widely used by industry for electromagnetic problems covering the entire frequency range, there are many situations where special methods are more effective. This is particularly the case for high￾frequency problems, e.g., millimeter and microwave integrated circuit structures where integral equation techniques and such procedures as transmission line modeling (TLM), spectral domain approach, method of lines, and wire grid methods are often preferred [Itoh, 1989] (see Chapter 43). Edge Elements Using potentials and nodal finite elements (see Fig. 44.1) rather than field components directly has the advantage that difficulties arising from field discontinuities at material interfaces can be avoided. However, if the element basis functions [see Eq. (44.5)] are expressed in terms of the field (H, say) constrained along an element edge, then tangential field continuity is enforced [Bossavit, 1988]. The field equations [Eq. (44.1)] in terms of the field intensity for the low frequency limit reduce to (44.7) FIGURE 44.1 Three-dimensional second-order Isoparametric finite elements. (a) Left, master tetrahedron, 10 nodes in local space (j, h, z); right, actual tetrahedron, 10 nodes in global space (x, y, z). (b) Left, master hexahedron, 20 nodes in local space (j, h, z); right, actual hexahedron, 10 nodes in global space (x, y, z). —¥—¥ + = H H s ¶ m ¶ ( ) t 0