Electromagnetics is a very important area in science and engineering, especially when the electromagnetic effects are coupled with mechanical and fluid flow systems. There are many important applications: electric motors, heating of furnaces/ovens, medical procedures, electromagnetic switches, electromagnetic pumps or brakes, wave guides, antennas, transmission lines, electromagnetic casting, nondestructive testing of metals, and so on.
All these electromagnetic phenomena and applications are uniformly governed by the general Maxwell's equations. For our multiphysics applications, we have therefore worked for some time to develop in the ADINA system a new modeling capability — the program ADINAEM — to solve the general Maxwell's equations with different loading and boundary conditions.
With the exciting new features provided by ADINAEM, the ADINA users can now solve the general Maxwell’s equations for many different problems and also couple the electromagnetic effects with fluid flows.
Fundamentally, the original firstorder Maxwell’s equations governing electromagnetics for the electric field intensity and the magnetic field intensity are, see Ref. [1],
with
Also, the Maxwell’s equations in the frequency domain (for harmonic analysis) are
where
In these equations, the electromagnetic material is characterized by , that is, the electric permittivity, magnetic permeability, and electric conductivity, respectively. The source terms are the two densities and , and the electric charge density . Together with appropriate boundary conditions, Maxwell's equations uniquely determine and in the problem domain.
In ADINAEM, two distinctly different formulations, namely a novel formulation and an formulation are used, where in the formulation as usual we use
For both formulations we utilize the finite element method. For efficiency and accuracy, instead of solving the firstorder Maxwell's equations, given above, we have reformulated these equations to secondorder relations, but without adding additional equations, see Ref. [2].
It is important to note that we offer in ADINAEM the two distinct formulations, that is, the formulation and the formulation. The reason is that the formulation is familiar to engineers and scientists and can therefore directly be used — but has the wellknown disadvantages. The formulation is novel, it uses the physical variables as unknowns, is more direct and these variables can directly be coupled to the actions of fluids and solids.
We should note as well that we do not use edgetype elements (with degrees of freedom at the element edges) but we use a more powerful formulation where — also — the finite element degrees of freedom directly couple to the usual fluid and solid elements used. The details of the formulation are presented in Ref. [2].
With our first release of ADINAEM, the following types of electromagnetic problems can be solved:
▪ Electrostatic fields  ▪ Magnetostatic fields  ▪ DC conduction 
▪ Timeharmonic  ▪ Eddy current  ▪ AC conduction 
▪ EM fields with Lorentz forces  ▪ EM fields coupled with temperature  ▪ Wave guide 
Of course, the pre and postprocessing for the ADINAEM models and solutions are performed using the ADINA User Interface.
Below we show the solutions of three example problems solved using ADINAEM.
Sharp material interface in harmonic analysis
In this first example — which is a good verification problem — we demonstrate the capability of ADINAEM in the calculation of electric and magnetic fields across a sharp material interface, with very different electromagnetic materials in the domains on each side. As shown in Figure 1, the material of the outside domain has zero conductivity while that of the inside domain has a very high conductivity. Because of these very different materials, the electric and magnetic fields have sharp variations across the material interface. Instead of using different formulations in the different domains, the problem is solved using ADINAEM with the formulation for both domains.
The plots in Figures 2 and 3 show the real and imaginary parts of the electric and magnetic field intensities.
Figure 1 Sharp interface problem: schematic
Figure 2 Sharp interface problem: vector plot of ; real part (left) and imaginary part (right)
Figure 3 Sharp interface problem: band plot of ; real part (left) and imaginary part (right)
We also compare the results obtained using ADINAEM with analytical results in Figures 4 and 5. The computational results agree closely with the theoretical values.
Figure 4 Sharp interface problem: , results from ADINA compared to analytical results; real part (left) and imaginary part (right)
Figure 5 Sharp interface problem: , results from ADINA compared to analytical results; real part (left) and imaginary part (right)
Electromagnetically induced mixing of glass melt in a pipe
This is a multiphysics electromagnetic stirring and mixing problem. The ADINAEM formulation and the ADINA CFD formulation are used, coupled, to
simulate the advective mixing in an electromagneticallydriven pipe
mixer.
The schematic of
this problem is as shown in Figure 6 below. In this example, fluid flows in a cylindrical tube subjected to stirring and mixing by the Lorentz force
generated by timedependent voltages in two electrodes that are immersed in
the conducting fluid, with the entire assembly in an otherwise externally imposed constant
magnetic field. Stirring and mixing occur in the plane perpendicular to
the flow direction due to the Lorentz force in that plane.
Figure 6 Electromagnetically induced mixing: schematic
The movie at the top shows the transient process of the mixing, starting from an inhomogeneous
concentration at the inlet. In Figures 7 to 9 below, we present a steadystate solution
of the electromagnetic mixing process, showing the calculated potentials
and ,
the velocity in a plane perpendicular to the main flow direction, and the mass concentrations at the inlet and outlet.
The homogeneous concentration at the outlet shows the perfect mixing achieved.
Figure 7 Electromagnetically induced mixing: Plot of (left) and (right)
Figure 8 Electromagnetically (chaotic) induced mixing: velocity vector plot near inlet
Figure 9 Electromagnetically induced mixing: mass ratio at inlet (left) and outlet (right)
Eddy current in a torus with cracks, induced by timeharmonic magnetic field
A schematic of this problem is shown in Figure 10 below. An eddy current is induced in a conductor by an externally
imposed harmonic magnetic flux. The toroid conductor has four cracks through its depth. These cracks modify the electric
and magnetic fields that would normally result were there no cracks, and this observation is the
basis of nondestructive testing (NDT) using electromagnetics. Only one
eighth of the whole domain is modeled. This 3D timeharmonic eddy
current problem is solved using the ADINAEM formulation. We
show, in Figures 11 and 12 below, the band plots of the real and imaginary parts of the electric and magnetic
field intensities. It can be seen that the cracks indeed change the direction and magnitude of both fields.
Figure 10 Eddy current in torus: schematic
Figure 11 Eddy current in torus: vector plot of ;
real part (left) and imaginary part (right)
Figure 12 Eddy current in torus: plot of ;
real part (left) and imaginary part (right)
For some other applications of ADINAEM, please see
 Multiphysics with Electromagnetics
 Microwave Heating
 Continuous Microwave Processing for Heating Materials
Clearly, the addition of ADINAEM to the ADINA system greatly extends and enhances the multiphysics capabilities offered in ADINA. The multiphysics capabilities can now be even more generally applied than before, with all the already existing powerful capabilities in ADINA, see here.
References

C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989.
 K. J. Bathe et al., The Direct Solution of Maxwell’s Equations in Multiphysics, in preparation.
Keywords:
Electromagnetics, Maxwell's Equations, multiphysics, fluid flow, electric field, electrostatic field, magnetic field, magnetostatic field, eddy current, wave guide,
nondestructive testing, NDT, Lorentz force, mixing