Finite element formulation for induction heating

The finite element formulation is used to solve for the heat source term used in the transient heat conduction equation.

The equations describing the electromagnetic phenomena can be found under electromagnetic field modeling. The finite element formulation is used to solve equation (19) on that page:

(19)

in order to get the heat source term, equation (21)

(21)

that is used by the existing cool solver to solve equation (22):

(22)

Since equation (19) is a time-harmonic simplified equation, the magnetic vector potential, , and the eddy currents, , must be expressed as complex numbers:

(23)

(24)

where and refer to a Cartesian component of the terms being solved, and the and subscripts to the real and imaginary components of the complex number.

The Galerkin approximation is applied to equation (19), separately, for each of the x, y and z components of the vector.

(25)

resulting in 3 equations of the following form:

(26)

where refers to the shape functions and to the volume of the element. Equation 26 is solved on each tetrahedral element for both the Real and Imaginary components of the unknown Cartesian components of the magnetic vector potential, . Hence has 6 degrees of freedom that need to be solved for.