The capability of modeling magnetic force coupling in current carrying solid conductors and solid magnetic materials exists in the following current-technology elements:

These elements use strong (matrix) coupling with volumetric integral Maxwell (Equation 5–140) or Lorentz force calculation methods.

In addition to the above elements, the following legacy elements support weak (load vector) coupling with surface integral Maxwell (Equation 5–136) force for ferromagnetic regions and Lorentz force for current carrying conductors:

This discussion addresses strong magnetic force coupling available with the PLANE223, SOLID226, and SOLID227 elements.

Magnetic solids and current carrying solid conductors deform when subject to a magnetic field. The magnetic body force {F^m} that causes the deformation can be derived from the Maxwell stress tensor [σ^M] (Landau and Lifshitz([355])):

(10–125)

where:

{H} = magnetic field intensity vector

{B} = magnetic flux density vector

As follows:

(10–126)

where:

[B] = strain-displacement matrix (see Equation 2–44)

vol = element volume

Applying the variational principle to the stress equation of motion and to the electromagnetic Equation 5–1 (Ampere’s law) coupled by magnetic force produces the following finite element equation:

(10–127)

where:

{u} = structural displacements (UX, UY, also UZ)

{A} = magnetic vector potential (2D) or edge-flux (3D) DOF (AZ)

[K] = element structural stiffness matrix (see [Ke] in Equation 2–58)

[M] = element mass matrix (see [Me] in Equation 2–58)

[KAA] = element magnetic reluctivity matrix (see [KAA] in Equation 5–112)

[C] = element structural damping matrix (discussed in Damping Matrices)

{F} = vector of nodal and surface forces (defined by Equation 2–56 and Equation 2–58)

{JS} = element source current density vector (see {JS} in Equation 5–112); applies only to structural-magnetic analysis (KEYOPT(1) = 10001)

{Jpm} = element remnant magnetization load vector (see {Jpm} in Equation 5–112)

The Maxwell force coupling method (KEYOPT(8) = 0) is applicable to both the magnetic solids and current carrying conductors. In the absence of other coupled-field effects that may affect the symmetry of Equation 10–127, the resulting system of magneto-structural equations coupled by the Maxwell force matrices is symmetric for KEYOPT(4) = 0, 1, and 2.

In the case of a current carrying conductor, structural and magnetic equations can be coupled by the Lorentz magnetic body force calculated from the electric current density {J} and the magnetic flux density {B} as follows:

(10–128)

where:

The system of structural and magnetic equations coupled by the Lorentz force coupling matrix is as follows:

(10–129)

where:

{NA} = vector of shape functions for {A}

The Lorentz force coupling method (KEYOPT(8) = 1) is applicable to current carrying conductors only and produces an unsymmetric system of equations (Equation 10–129).

The finite element equations for structural-electromagnetic (KEYOPT(1) = 10101) and structural-stranded coil (KEYOPT(1) = 10201) analyses are the combinations of Equation 10–127 or Equation 10–129 with Equation 5–112 and Equation 5–162, respectively.

Not that for a conducting solid (KEYOPT(1) = 10101), the electric current density {J} consists of the DC {J_s}, eddy {J_e}, and velocity {J_v} current components (Equation 5–123):

(10–130)

where:

[σ] = electrical conductivity matrix (inverse of the electrical resistivity matrix input as RSVX, RSVY, RSVZ on the MP command)

V = electric scalar potential (VOLT)

{v} = velocity vector =

Although the eddy and velocity current contributions to the magnetic force acting on a conductor were illustrated using the Lorentz force method, these effects are included with the Maxwell force method as well when it is applied to current carrying conductors.