TEAM 4: Eddy current simulation of a conducting brick in a decaying magnetic field

This benchmark problem verifies the case of a rectangular brick with a rectangular hole through the center placed in a uniform magnetic field that is decaying in time. It is part of the standard suite of problems designed for Testing Electromagnetic Analysis Methods (TEAM). The objective is to compute the circulating eddy currents induced in the brick and the ensuing Joule heat dissipated due to the magnetic field that is decaying in time.

This page discusses:

ProductsAbaqus/Standard

Problem description

The problem setup is shown in Figure 1, which depicts a conductive rectangular brick with a rectangular hole placed in a uniform magnetic field that is decaying in time. The dimensions of the brick are a = 0.1524 m, b = 0.1016 m, and c = 0.0508 m. The brick is assumed to be made of an aluminum alloy with a resistivity of ρ = 3.94 × 10–8Ωm and a relative magnetic permeability of μr = 1.0. The rectangular hole is centered in and penetrates through the brick. The dimensions of the hole are assumed to be l = 0.0889 m and w = 0.0381 m. The orientation of the uniform magnetic field is parallel to the direction of penetration of the hole and is assumed to be decaying as Bz=B0⁢exp⁡(-t/τ), where B0 = 0.1 T and τ = 0.0119 s. The medium surrounding the brick is assumed to have properties similar to that of a vacuum.

Model and boundary conditions

The magnetic vector potential formulation is used to solve this problem. Due to the symmetry of the problem, it is sufficient to model the first octant of the problem domain. Appropriate boundary conditions are imposed on the symmetry planes x = 0, y = 0, and z = 0. Due to the symmetry of the external magnetic field with respect to the planes x = 0 and y = 0, the magnetic vector potential is normal to these symmetry planes, which is enforced by a homogeneous Dirichlet boundary condition. Similarly, due to the asymmetry of the external magnetic field with respect to the plane z = 0, the magnetic flux density is normal to the symmetry plane, which is enforced by a homogeneous Neumann boundary condition.

The presence of the conducting brick in a time-varying magnetic field generates eddy currents in the brick, which in turn generate their own magnetic field in the vicinity of the brick. The magnetic field far away from the brick is not changed significantly by these eddy currents from that of the external magnetic field. Since the external magnetic field is pointing in the z-direction, it is easier to specify the far-field boundary conditions if we choose planar truncation boundaries that are parallel to the planes x = 0, y = 0, and z = 0. The external magnetic field on the truncation boundaries is specified as a surface current density load.

Results and discussion

Figure 2 shows the z-component of the magnetic flux density, Bz, computed using an Abaqus/Standard low-frequency transient electromagnetic analysis performed over a period of 20 ms. The magnetic field is plotted along the positive z-axis of the model starting from the center of the hole. The curves in the plot correspond to the field values at different analysis times, as indicated in the legend. The axes are scaled to facilitate the comparison of the results against the published results in Kameari (1988). The horizontal axis is multiplied by 6.35 to convert the true distance along the z-axis to millimeters, and the vertical axis is multiplied by 0.01 to convert the flux density to Tesla. The plot indicates that far away from the conducting brick the magnetic field is the same as that of the external magnetic field, which is decaying in time. However, at the center of the brick the magnetic field is larger due to the eddy currents induced in the brick; these currents try to compensate for the magnetic field that is reducing in time. The results compare very well with those published in Kameari.

Figure 3 shows the total induced current that is flowing across the cross-section of the conducting brick plotted against the analysis time. The total current is obtained by integrating the current density output across the cross-section. The results compare very well to those published in Kameari.

Input files

team4_emc3d8.inp

Low-frequency transient eddy current analysis of an aluminum brick with a hole placed in a uniform magnetic field that is decaying in time; modeled using EMC3D8 elements and symmetry boundary conditions.

References

  1. Kameari,  A., “Results for Benchmark Calculations of Problem 4 (the FELIX Brick),” COMPEL, vol. 7, pp. 65–80, 1988.

Figures

Figure 1. Geometry of a rectangular brick with a hole placed in a decaying magnetic field.

Figure 2. Magnetic flux density along the z-axis.

Figure 3. Total induced current flowing across the cross-section of the conducting brick.