Any direct heater can be utilized as an inductor of the traveling magnetic
field in *CrysMAS*, see
TMF effect
for details.

The numerical model of traveling magnetic field supports only the computation of the Lorentz force in the electro conducting fluid (melt). No application of the phase shifted inductors to the Joule heat generation is available.

The given distribution of electric currents is represented by features of corresponding heaters with the TMF effect. Electric currents in heaters produces sources of the electromagnetic field. Currents are assumed to be harmonic and have the same circular frequency.

For computation of the produced Lorentz forces the vector potential formulation is applied to the Maxwell equations. The vector potential is assumed to have only the axial component in the axisymmetric coordinate system. The corresponding eddy current density in the electric conductor has only azimuthal component. The Lorentz force is computed as a cross-product of the induced eddy current with the magnetic field intensity. The magnetic field intensity is found by a rotation operator applied to the vector potential field.

Vector potential is represented by two harmonic components, one of which oscillates in phase with the primary current in the inductor and another is shifted by 90 degrees. Both components are computed as a solution of the coupled differential partial equations for the boundary value problem. The mathematical formulation for the single inductor is exactly the same as the model of alternating magnetic fields (AMF), see section Physical background for more details. The TMF model can be considered as an extension of the AMF model, whereby inductors may have currents of arbitrary phase.

It should be emphasized that the same rules should be satisfied regarding the skin effect in the electric conductors and its numerical resolution.

In the TMF model many heaters with different phases of the electric current should be present. Then the algorithm separates all inductors into groups of inductors with the same phase shift. The vector potential components are computed then independently for each inductor group associated with the particular phase shift as if no other groups of inductors with another current phase are present. The results for the components of the vector potential are stored for each group of inductors.

Then the Lorentz force is computed using the stored vector potentials of all groups. It is assumed that the resulted vector potential is a superposition of fields created by each group of inductors. Therefore for each node the components of the vector potential of different phase are summed as complex numbers. The real and imaginary parts of the sum are then the computed in-phase and out-of-phase component of the resulted added up vector potential.

The Lorentz force is computed from the added up vector potential in the same manner as described for the AMF model with inductor currents of the same phase.

The potential equations are solved like equations of the AMF model only on the unstructured mesh. The following computation of the Lorentz force is succeeded again on the unstructured mesh. If the Lorentz force is utilized on the structured mesh, then it will be transferred automatically from the overlapped unstructured mesh to the second layer of the structured mesh. But the numerical resolution and treatment of the skin effect in the melt volume is defined by quality of the unstructured mesh.

Provided electric conductivities and relative magnetic permeabilities of all materials in the setup are not temperature dependent, the vector potential required for the Lorentz force is computed one from the linear equations system. The equations system is solved for the total computational domain, but thanks to their linearity no repeated iterations should be done. Therefore the computation takes only a short time (seconds).