The analytical solution of the Biot-Savart equation can be complex in some cases, and its numerical integration is commonly more appropriate. In this paper, it is integrated using the Gauss-Legendre method through 1, 2 and 3-D domains, using first and second-order (curvilinear) isoparametric mapping. In order to verify the gain of accuracy with second-order elements, the results obtained are compared with analytical cases and with the Finite Element Method. Then this paper presents an adaptive method which prots from the accuracy along those elements with higher energy values, by reducing the number of Gauss points along the elements with lower energy. This approach reduces the total number of Gauss points evaluated during the integration process and provides a possibility to choose an interesting trade-off between simulation time and accuracy.
1. Azpurua, M. A., "A semi-analytical method for the design of coil-systems for homogeneous magnetostatic field generation," Progress In Electromagnetics Research B, Vol. 37, 171-189, 2012. doi:10.2528/PIERB11102606
2. Bastos, J. P. A. and N. Sadowski, Electromagnetic Modeling by Finite Element Methods, Marcel Dekker, New York, 2003. doi:10.1201/9780203911174
3. Carley, M., "Evaluation of Biot-Savart integrals on tetrahedral meshes," SIAM Journal on Scientific Computing, 2017.
4. Ciric, I. R., "Simple analytical expressions for the magnetic field of current coils," IEEE Transactions on Magnetics, Vol. 27, No. 1, 669-673, 1991. doi:10.1109/20.101115
5. Dular, P., "Modfielisation du champ magnfietique et des courants induits dans des systémes tridimensionnels non linéaires,", Ph. d, Université de Liege, 1996.
6. Dular, P., C. Geuzaine, F. Henrotte, and W. Legros, "A general environment for the treatment of discrete problems and its application to the finite element method," IEEE Transactions on Magnetics, Vol. 34, No. 5, 3395-3398, September 1998. doi:10.1109/20.717799
7. Dular, P., L. Krähenbühl, M. V. Ferreira da Luz, P. Kuo-Peng, and C. Geuzaine, "Progressive inductor modeling via a finite element subproblem method," COMPEL, Vol. 34, No. 3, 851-863, May 2015. doi:10.1108/COMPEL-10-2014-0279
8. Geuzaine, C. and J.-F. Remacle, "Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities," International Journal for Numerical Methods in Engineering, Vol. 79, No. 11, 1309-1331, September 2009. doi:10.1002/nme.2579
9. Guibert, A., J. L. Coulomb, O. Chadebec, and C. Rannou, "A post-processing integral formulation for the computation of magnetic field in conductors," IEEE Transactions on Magnetics, Vol. 47, No. 5, 1334-1337, 2011. doi:10.1109/TMAG.2010.2102341
10. Gyimesi, M. G., D. Lavers, T. Pawlak, and D. Ostergaard, "Biot-savart integration for bars and Arcs," IEEE Transactions on Magnetics, Vol. 29, No. 6, 2389-2391, 1993. doi:10.1109/20.281007
11. Lee, H.-B. and H.-J. Song, "Efficient magnetic field calculation method for pancake coil using biot-savart law," 2006 12th Biennial IEEE Conference on Electromagnetic Field Computation, Vol. 27, 193, IEEE, 2006.
13. Jackson, J. D., Classical Electrodynamics, 3rd Ed., John Wiley & Sons, New York, 1999.
14. Jin, J.-M., The Finite Element Method in Electromagnetics, 2nd Ed., A Wiley-Interscience Publication, Wiley, New York, 2002.
15. Kalhor, H. A., "Comparison of Ampere’s circuital law (ACL) and the law of Biot-Savart (LBS)," IEEE Transactions on Education, Vol. 31, No. 3, 236-238, 1988. doi:10.1109/13.2322
16. Kim, K. C. and J. Lee, "Comparison of Biot Savart simulation and 3D finite element simulation of the electromagnetic forces acting on end windings of electrical machines," 12th Biennial IEEE Conference on Electromagnetic Field Computation, CEFC 2006, Vol. 135, No. 6, 4244, 2006.
17. Landini, M., "About the physical reality of ‘maxwell’s displacement current’ in classical electrodynamics," Progress In Electromagnetics Research, Vol. 144, 329-343, 2014. doi:10.2528/PIER13111501
18. Le-Duc, T., O. Chadebec, J.-M. Guichon, and G. Meunier, "New coupling between PEEC method and an integro-differential approach for modeling solid conductors in the presence of magneticconductive thin plates," IET 8th International Conference on Computation in Electromagnetics (CEM 2011), Vol. 19, 30-31, 2011. doi:10.1049/cp.2011.0023
19. Le-Van, V., "Développement de formulations intégrales de volume en magnétostatique,", Ph.D, Université Grenoble Alpes - Laboratoire de Génie Electrique de Grenoble, Grenoble, 2015.
20. Meunier, G., The Finite Element Method for Electromagnetic Modeling, John Wiley & Sons, Inc., Hoboken, 2008. doi:10.1002/9780470611173
21. Modric, T., S. Vujevic, and D. Lovric, "3D computation of the power lines magnetic field," Progress In Electromagnetics Research M, Vol. 41, 1-9, 2015. doi:10.2528/PIERM14122301
22. Nunes, A. S., O. Chadebec, P. Kuo-Peng, P. Dular, and G. Meunier, "A coupling between the facet finite element and reluctance network methods in 3-D," IEEE Transactions on Magnetics, Vol. 53, No. 10, 1-10, October 2017. doi:10.1109/TMAG.2017.2723576
23. Suh, J.-C., "The evaluation of the biot-savart integral," Journal of Engineering Mathematics, Vol. 37, No. 4, 375-395, 2000. doi:10.1023/A:1004666000020
24. Urankar, L., "Vector potential and magnetic field od current-carrying finite arc segment in analytical form, Part III: Exact computation for rectangular cross section," IEEE Transactions on Magnetics, Vol. 18, No. 6, 1860-1867, 1982. doi:10.1109/TMAG.1982.1062166
25. Urankar, L., "Vector potential and magnetic field of current-carying finite arc segment in analytical form, part II: Thin sheet approximation," IEEE Transactions on Magnetics, Vol. 18, No. 3, 911-917, May 1982. doi:10.1109/TMAG.1982.1061927
26. Urankar, L., "Vector potential and magnetic field of current-carrying finite arc segment in analytical form, part IV: General three-dimensional current density," IEEE Trans. Magn., Vol. 20, No. 6, 2145-2150, November 1984. doi:10.1109/TMAG.1984.1063579
27. Urankar, L., "Vector potential and magnetic field of current-carrying circular finite arc segment in analytical form - Part V. Polygon cross section," IEEE Transactions on Magnetics, Vol. 26, No. 3, 1171-1180, May 1990. doi:10.1109/20.53995
28. Urankar, L. and P. Henninger, "Compact extended algorithms for elliptic integrals in electromagnetic field and potential computations. I. Elliptic integrals of the first and second kind with extended integration range," IEEE Transactions on Magnetics, Vol. 27, No. 5, 4338-4342, September 1991. doi:10.1109/20.105059
29. Volkmar, C., T. Baruth, J. Simon, U. Ricklefs, and R. Thueringer, "Arbitrarily shaped coils’ inductance simulation based on a 3-dimensional solution of the Biot-Savart law," Proceedings of the International Spring Seminar on Electronics Technology, No. 1, 210-215, 2013. doi:10.1109/ISSE.2013.6648244
30. Weggel, C. F. and D. P. Schwartz, "New analytical formulas for calculating magnetic field," IEEE Transactions on Magnetics, Vol. 24, No. 2, 1544-1547, March 1988. doi:10.1109/20.11540
31. Wilton, D., S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, "Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains," IEEE Transactions on Antennas and Propagation, Vol. 32, No. 3, 276-281, March 1984. doi:10.1109/TAP.1984.1143304
32. Wu, J.-Z., H.-Y. Ma, and M.-D. Zhou, Vorticity and Vortex Dynamics, Springer Berlin Heidelberg, 2007.