VM-LSDYNA-EMAG-009
VM-LSDYNA-EMAG-009
Hollow Conducting Sphere in a Uniform Magnetic Field (TEAM 6)
Overview
| Reference: | Emson, C.R.I. (1988). Results for a hollow sphere in uniform field (Benchmark Problem 6). COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 7, 89-101. |
| Analysis Type(s): | Electromagnetism |
| Element Type(s): | Solid Elements ELFORM 1 |
| Input Files: | Link to Input Files Download Page |
Test Case
The TEAM 6 problem is a hollow conducting sphere with an inner radius of 5 cm and an outer radius of 5.5 cm. The sphere has a conductivity of 5.0 x 10E08 S/m, and a relative permeability of unity. A uniform field is incident (in the z-direction) and oscillates at 50 Hz. It varies sinusoidally with time. The goal is to determine the real and imaginary values of the magnetic field along a given axis. Results are then compared to experimental results in the reference.
| Material Properties | Geometric Properties | Loading |
|---|---|---|
 |  |  |
 |  |  |
Analysis Assumptions and Modeling Notes
The LS-DYNA *EM_CONTROL card is set to 4 to activate the Frequency-based Eddy Current solver. The external magnetic field is applied using the *EM_EXTERNAL_FIELD card which points to a load curve with the magnetic field value.
The mesh is formed by a solid mesh with element formulation 1, which is a constant stress solid element. A shell mesh is created on top of the solid one with element formulation 2. The shell mesh is used by the BEM (Boundary Element Method) solver.
Results Comparison
The analytic solution is compared with LS-DYNA output. Simulation results are extracted from em_pointout.dat and em_pointout_phi.dat. The real and imaginary parts of the magnetic field are calculated using the expressions below:
 | (31) |
 | (32) |
Along with the simulation and analytical solutions, an orange-filled area represents the absolute error. You can see that the real part error is always within 0.05 T absolute error, and the imaginary part error is within 0.01 T. This indicates a strong correlation between both analytical and simulation results.
The error between the analytical solution and the simulation is presented in the tables below. They show that there is a strong correlation between the reference and the simulation.
Table 15: z-Bfield (real) for several x coordinates (at y = 0, z = 0) for 50 Hz
| Result | Target (T) | Workbench LS-DYNA (T) | Error (T) |
|---|---|---|---|
| zBfield (real) for Z = Y = 0 mm, X = 0 mm | -0.036827 | -0.031609 | 0.005218 |
| zBfield (real) for Z = Y = 0 mm, X = 40 mm | -0.037803 | -0.030525 | 0.007278 |
| zBfield (real) for Z = Y = 0 mm, X = 45 mm | -0.038779 | -0.030156 | 0.008623 |
| zBfield (real) for Z = Y = 0 mm, X = 47 mm | -0.039755 | -0.029996 | 0.009759 |
| zBfield (real) for Z = Y = 0 mm, X = 50 mm | -0.040731 | 0.046344 | 0.087075 |
| zBfield (real) for Z = Y = 0 mm, X = 51 mm | 0.141575 | 0.197926 | 0.056351 |
| zBfield (real) for Z = Y = 0 mm, X = 52 mm | 0.35261 | 0.366465 | 0.013855 |
| zBfield (real) for Z = Y = 0 mm, X = 53 mm | 0.654385 | 0.785353 | 0.130968 |
| zBfield (real) for Z = Y = 0 mm, X = 54 mm | 1.020525 | 1.038617 | 0.018092 |
| zBfield (real) for Z = Y = 0 mm, X = 55 mm | 1.41885 | 1.311444 | 0.107406 |
| zBfield (real) for Z = Y = 0 mm, X = 57 mm | 1.40115 | 1.410479 | 0.009329 |
| zBfield (real) for Z = Y = 0 mm, X = 60 mm | 1.354 | 1.351982 | 0.002018 |
| zBfield (real) for Z = Y = 0 mm, X = 65 mm | 1.27025 | 1.27698 | 0.00673 |
| zBfield (real) for Z = Y = 0 mm, X = 70 mm | 1.2178 | 1.22185 | 0.00405 |
| zBfield (real) for Z = Y = 0 mm, X = 75 mm | 1.1772 | 1.180425 | 0.003225 |
| zBfield (real) for Z = Y = 0 mm, X = 80 mm | 1.14165 | 1.1487 | 0.00705 |
| zBfield (real) for Z = Y = 0 mm, X = 85 mm | 1.11475 | 1.123994 | 0.009244 |
| zBfield (real) for Z = Y = 0 mm, X = 90 mm | 1.097 | 1.104471 | 0.007471 |
| zBfield (real) for Z = Y = 0 mm, X = 95 mm | 1.07915 | 1.088839 | 0.009689 |
| zBfield (real) for Z = Y = 0 mm, X = 100 mm | 1.0663 | 1.076175 | 0.009875 |
Table 16: z-Bfield (imaginary) for several x coordinates (at y = 0, z = 0) for 50 Hz
| Result | Target (T) | Workbench LS-DYNA (T) | Error (T) |
|---|---|---|---|
| zBfield (imaginary) for Z = Y = 0 mm, X = 0 mm | -0.036827 | -0.04051 | 0.003683 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 40 mm | -0.039417 | -0.040261 | 0.000844 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 45 mm | -0.042007 | -0.040178 | 0.001828 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 47 mm | -0.044596 | -0.040143 | 0.004453 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 50 mm | -0.047186 | -0.101909 | 0.054723 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 51 mm | -0.14735 | -0.217861 | 0.070511 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 52 mm | -0.323775 | -0.310614 | 0.013161 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 53 mm | -0.350825 | -0.359903 | 0.009078 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 54 mm | -0.297115 | -0.277385 | 0.01973 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 55 mm | -0.051856 | -0.093669 | 0.041813 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 57 mm | 0.030697 | 0.034155 | 0.003458 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 60 mm | 0.025557 | 0.029297 | 0.00374 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 65 mm | 0.018718 | 0.023066 | 0.004347 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 70 mm | 0.017717 | 0.018482 | 0.000765 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 75 mm | 0.011706 | 0.015036 | 0.00333 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 80 mm | 0.008396 | 0.012396 | 0.004 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 85 mm | 0.002777 | 0.010339 | 0.007562 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 90 mm | 0.001489 | 0.008712 | 0.007223 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 95 mm | -0.000708 | 0.00741 | 0.008117 |
| zBfield (imaginary) for Z = Y = 0 mm, X = 100 mm | -0.002559 | 0.006354 | 0.008913 |
Table 17: z-Bfield (real) for several z coordinates (at y = 0, x = 0) for 50 Hz
| Result | Target (T) | Workbench LS-DYNA (T) | Error (T) |
|---|---|---|---|
| zBfield (real) for X = Y = 0 mm, Z = 0 mm | -0.035482 | -0.031609 | 0.003873 |
| zBfield (real) for X = Y = 0 mm, Z = 40 mm | -0.032735 | -0.031099 | 0.001636 |
| zBfield (real) for X = Y = 0 mm, Z = 45 mm | -0.032735 | -0.030809 | 0.001926 |
| zBfield (real) for X = Y = 0 mm, Z = 47 mm | -0.032845 | -0.030673 | 0.002172 |
| zBfield (real) for X = Y = 0 mm, Z = 50 mm | -0.032378 | -0.034773 | 0.002395 |
| zBfield (real) for X = Y = 0 mm, Z = 51 mm | -0.029137 | -0.030192 | 0.001054 |
| zBfield (real) for X = Y = 0 mm, Z = 52 mm | -0.013261 | -0.021394 | 0.008134 |
| zBfield (real) for X = Y = 0 mm, Z = 53 mm | -0.001453 | 0.010449 | 0.011902 |
| zBfield (real) for X = Y = 0 mm, Z = 54 mm | 0.024839 | 0.034511 | 0.009672 |
| zBfield (real) for X = Y = 0 mm, Z = 55 mm | 0.073692 | 0.064494 | 0.009198 |
| zBfield (real) for X = Y = 0 mm, Z = 57 mm | 0.172485 | 0.17749 | 0.005005 |
| zBfield (real) for X = Y = 0 mm, Z = 60 mm | 0.291493 | 0.294833 | 0.00334 |
| zBfield (real) for X = Y = 0 mm, Z = 65 mm | 0.441277 | 0.445238 | 0.00396 |
| zBfield (real) for X = Y = 0 mm, Z = 70 mm | 0.561872 | 0.555752 | 0.00612 |
| zBfield (real) for X = Y = 0 mm, Z = 75 mm | 0.640226 | 0.638766 | 0.001459 |
| zBfield (real) for X = Y = 0 mm, Z = 80 mm | 0.709878 | 0.702327 | 0.007552 |
| zBfield (real) for X = Y = 0 mm, Z = 85 mm | 0.755552 | 0.751812 | 0.00374 |
| zBfield (real) for X = Y = 0 mm, Z = 90 mm | 0.795653 | 0.790911 | 0.004742 |
| zBfield (real) for X = Y = 0 mm, Z = 95 mm | 0.833638 | 0.822211 | 0.011427 |
| zBfield (real) for X = Y = 0 mm, Z = 100 mm | 0.854896 | 0.847564 | 0.007332 |
Table 18: z-Bfield (imaginary) for several z coordinates (at y = 0, x = 0) for 50 Hz
| Result | Target (T) | Workbench LS-DYNA (T) | Error (T) |
|---|---|---|---|
| zBfield (imaginary) for X = Y = 0 mm, Z = 0 mm | -0.040886 | -0.04051 | 0.000376 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 40 mm | -0.040886 | -0.040368 | 0.000518 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 45 mm | -0.040886 | -0.040284 | 0.000602 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 47 mm | -0.040886 | -0.040246 | 0.00064 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 50 mm | -0.040886 | -0.041651 | 0.000765 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 51 mm | -0.040886 | -0.045082 | 0.004196 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 52 mm | -0.051165 | -0.051208 | 0.000044 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 53 mm | -0.060888 | -0.0671 | 0.006213 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 54 mm | -0.068386 | -0.073624 | 0.005238 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 55 mm | -0.070876 | -0.076305 | 0.005429 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 57 mm | -0.064156 | -0.068527 | 0.004371 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 60 mm | -0.055779 | -0.058769 | 0.00299 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 65 mm | -0.042046 | -0.046254 | 0.004208 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 70 mm | -0.032461 | -0.037051 | 0.004591 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 75 mm | -0.025484 | -0.030135 | 0.004651 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 80 mm | -0.019717 | -0.024837 | 0.005121 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 85 mm | -0.014993 | -0.020711 | 0.005718 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 90 mm | -0.01134 | -0.01745 | 0.00611 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 95 mm | -0.007632 | -0.014839 | 0.007207 |
| zBfield (imaginary) for X = Y = 0 mm, Z = 100 mm | -0.003581 | -0.012724 | 0.009143 |