Electromagnetics with ADINA

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, non-destructive 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 ADINA-EM — to solve the general Maxwell's equations with different loading and boundary conditions.

With the exciting new features provided by ADINA-EM, 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 first-order 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 ADINA-EM, 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 first-order Maxwell's equations, given above, we have reformulated these equations to second-order relations, but without adding additional equations, see Ref. [2].

It is important to note that we offer in ADINA-EM 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 well-known 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 edge-type 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 ADINA-EM, the following types of electromagnetic problems can be solved:

Of course, the pre- and post-processing for the ADINA-EM models and solutions are performed using the ADINA User Interface.

Below we show the solutions of three example problems solved using ADINA-EM.

Sharp material interface in harmonic analysis

In this first example — which is a good verification problem — we demonstrate the capability of ADINA-EM 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 ADINA-EM 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 ADINA-EM 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 ADINA-EM formulation and the ADINA CFD formulation are used, coupled, to simulate the advective mixing in an electromagnetically-driven 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 time-dependent 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 steady-state 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 time-harmonic 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 non-destructive testing (NDT) using electromagnetics. Only one eighth of the whole domain is modeled. This 3D time-harmonic eddy current problem is solved using the ADINA-EM 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 ADINA-EM, please see

• Multiphysics with Electromagnetics

• Microwave Heating

• Continuous Microwave Processing for Heating Materials

Clearly, the addition of ADINA-EM 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

1. C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989.

2. 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, non-destructive testing, NDT, Lorentz force, mixing