Finite element formulation of the general magnetostatic problem in the space of solenoidal vector functions
Author:
Mark J. Friedman
Journal:
Math. Comp. 43 (1984), 415431
MSC:
Primary 78A30; Secondary 65N30, 7808
MathSciNet review:
758191
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A new finite element method for the solution of the general magnetostatic problem is formulated and analyzed. The space of trial functions consists of solenoidal piecewise polynomial vector functions. We start with an integral formulation, in terms of the flux density, in the domain occupied by magnetic material. Using the properties [10] of the spectrum of the relevant singular integral operator we derive a weak formulation involving an integral operator on the boundary only. Thus the resulting finite element matrix consists of a sparse part corresponding to the interior of the iron domain and a full part corresponding to the boundary. Using the method of monotone operators, existence and uniqueness of the solution of the weak formulation as well as its discretization are proven. Error estimates are derived with the special emphasis on the case when magnetic permeability is large. Finally, solution of the problem by successive iteration is analyzed.
 [1]
P. G. Akishin, S. B. Vorozhtsoz & E. P. Zhidkov, Calculation of the Magnetic Field of the Isochronous Cyclotron Sector Magnet by the Integral Equation Method, Proc. Compumag Conf., Grenoble, 1978.
 [2]
A. G. Armstrong, A. M. Collie, C. J. Diserens, N. J. Newman, M. Simkin & C. W. Trowbridge, New Developments in the Magnet Design Program GFUN, Rutherford Laboratory Report No. RL5060.
 [3]
J. H. Bramble & J. E. Pasciak, "A new computational approach for the linearized scalar potential formulation of the magnetostatic field problem," IEEE Trans. Mag., v. Mag18, 1982, pp. 357361.
 [4]
M.
V. K. Chari and P.
P. Silvester (eds.), Finite elements in electrical and magnetic
field problems, John Wiley & Sons, Ltd., Chichester, 1980. Wiley
Series in Numerical Methods in Engineering; A WileyInterscience
Publication. MR
589746 (81j:78009)
 [5]
Mark
J. Friedman, Mathematical study of the nonlinear singular integral
magnetic field equation. I, SIAM J. Appl. Math. 39
(1980), no. 1, 14–20. MR 585825
(81m:78006), http://dx.doi.org/10.1137/0139003
 [6]
Mark
J. Friedman, Mathematical study of the nonlinear singular integral
magnetic field equation. II, SIAM J. Numer. Anal. 18
(1981), no. 4, 644–653. MR 622700
(83i:78006a), http://dx.doi.org/10.1137/0718042
 [7]
Mark
J. Friedman, Mathematical study of the nonlinear singular integral
magnetic field equation. III, SIAM J. Math. Anal. 12
(1981), no. 4, 536–540. MR 617712
(83i:78006b), http://dx.doi.org/10.1137/0512046
 [8]
M. J. Friedman & J. S. Colonias, "On the coupled differentialintegral equations for the solution of the general magnetostatic problem," IEEE Trans. Mag., v. Mag18, No. 2, March 1982, pp. 336339.
 [9]
M. J. Friedman, Finite Element Formulation of the General Magnetostatic Problem in the Space of Solenoidal Vector Functions, Ph.D. Thesis, Cornell University, 1982.
 [10]
Mark
J. Friedman and Joseph
E. Pasciak, Spectral properties for the
magnetization integral operator, Math.
Comp. 43 (1984), no. 168, 447–453. MR 758193
(86f:78007), http://dx.doi.org/10.1090/S00255718198407581931
 [11]
R. Glowinski & A. Marrocco, "Numerical solution of twodimensional magnetostatic problems by augmented Lagrangian methods," Comput. Methods Appl. Mech. Engrg., v. 12, 1977, pp. 3346.
 [12]
F.
Hecht, Construction d’une base de fonctions 𝑃₁
non conforme à divergence nulle dans 𝑅³, RAIRO
Anal. Numér. 15 (1981), no. 2, 119–150
(French, with English summary). MR 618819
(83j:65023)
 [13]
J. L. Lions & E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, SpringerVerlag, New York, 1972.
 [14]
M. H. Lean & A. Wexler, Accurate Field Computation with the Boundary Element Method, Proc. Compumag Conf., Chicago, 1982.
 [15]
B. H. McDonald & W. Wexler, "Mutually constrained partial differential and integral equation field formulation," in Finite Elements in Electrical and Magnetic Field Problems (M. V. U. Chari and P. Silvester, eds.), Wiley, New York, 1978.
 [16]
I.
D. Maergoĭz, Iteratsionnye metody rascheta staticheskikh
polei v neodnorodnykh, anizotropnykh i nelineinykh sredakh,
“Naukova Dumka”, Kiev, 1979 (Russian). MR 533743
(80d:78021)
 [17]
I. D. Mayergoyz, "On numerical investigation of magnetostatic field in nonlinear ferromagnetic media," Sbornik Kibernetika, Vychislitel'naya Tekhnika, v. 17, 1972. (Russian)
 [18]
J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.
 [19]
J. C. Nedelec, Mixed Finite Elements in , Rapport Interne no. 49, Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau, 1979.
 [20]
Joseph
E. Pasciak, An iterative algorithm for the volume integral method
for magnetostatics problems, Comput. Math. Appl. 8
(1982), no. 4, 283–290. MR 679401
(83k:78012), http://dx.doi.org/10.1016/08981221(82)900104
 [21]
Joseph
E. Pasciak, A new scalar potential formulation of
the magnetostatic field problem, Math.
Comp. 43 (1984), no. 168, 433–445. MR 758192
(86f:78009), http://dx.doi.org/10.1090/S0025571819840758192X
 [22]
I. I. Pekker, "Calculation of magnetic systems by integration over field sources," Izv. Vyssh. Uchebn. Zaved. Electromekh., v. 9, 1964, pp. 10471051 (Russian)
 [23]
Walter
Rudin, Principles of mathematical analysis, 3rd ed.,
McGrawHill Book Co., New YorkAucklandDüsseldorf, 1976.
International Series in Pure and Applied Mathematics. MR 0385023
(52 #5893)
 [24]
J. Simkin & C. W. Trowbridge, "On the use of total scalar potential in the numerical solution of field problems in electromagnetics," Internat. J. Numer. Methods Engrg., v. 14, 1979, pp. 423440.
 [25]
J. Simkin & C. W. Trowbridge, ThreeDimensional Computer Program (TOSCA) for Nonlinear Electromagnetic Fields, Rutherford Laboratory Report No. RL79097.
 [26]
R. Temam, On the Theory and Numerical Analysis of the Navier Stokes Equations, NorthHolland, Amsterdam, 1977.
 [27]
M. M. Vaĭnberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Halsted Press, New YorkToronto, 1973.
 [1]
 P. G. Akishin, S. B. Vorozhtsoz & E. P. Zhidkov, Calculation of the Magnetic Field of the Isochronous Cyclotron Sector Magnet by the Integral Equation Method, Proc. Compumag Conf., Grenoble, 1978.
 [2]
 A. G. Armstrong, A. M. Collie, C. J. Diserens, N. J. Newman, M. Simkin & C. W. Trowbridge, New Developments in the Magnet Design Program GFUN, Rutherford Laboratory Report No. RL5060.
 [3]
 J. H. Bramble & J. E. Pasciak, "A new computational approach for the linearized scalar potential formulation of the magnetostatic field problem," IEEE Trans. Mag., v. Mag18, 1982, pp. 357361.
 [4]
 M. V. K. Chari & P. P. Silvester (editors), Finite Elements in Electrical and Magnetic Field Problems, Wiley, New York, 1978. MR 589746 (81j:78009)
 [5]
 M. J. Friedman, "Mathematical study of the nonlinear singular integral magnetic field equation 1," SIAM J. Appl. Math., v. 44, 1980, pp. 1420. MR 585825 (81m:78006)
 [6]
 M. J. Friedman, "Mathematical study of the nonlinear singular integral magnetic field equation 2," SIAM J. Numer. Anal., v. 18, 1981, pp. 644653. MR 622700 (83i:78006a)
 [7]
 M. J. Friedman, "Mathematical study of the nonlinear singular integral magnetic field equation 3," SIAM J. Math. Anal., v. 12, 1981, pp. 536540. MR 617712 (83i:78006b)
 [8]
 M. J. Friedman & J. S. Colonias, "On the coupled differentialintegral equations for the solution of the general magnetostatic problem," IEEE Trans. Mag., v. Mag18, No. 2, March 1982, pp. 336339.
 [9]
 M. J. Friedman, Finite Element Formulation of the General Magnetostatic Problem in the Space of Solenoidal Vector Functions, Ph.D. Thesis, Cornell University, 1982.
 [10]
 M. J. Friedman & J. E. Pasciak, "Spectral properties for the magnetization integral operator," Math. Comp., v. 43, 1984, pp. 447453. MR 758193 (86f:78007)
 [11]
 R. Glowinski & A. Marrocco, "Numerical solution of twodimensional magnetostatic problems by augmented Lagrangian methods," Comput. Methods Appl. Mech. Engrg., v. 12, 1977, pp. 3346.
 [12]
 F. Hecht, "Construction d'une base de fonctions non conforme à divergence nulle dans ," RAIRO Anal. Numér., v. 15, 1981, pp. 119150. MR 618819 (83j:65023)
 [13]
 J. L. Lions & E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, SpringerVerlag, New York, 1972.
 [14]
 M. H. Lean & A. Wexler, Accurate Field Computation with the Boundary Element Method, Proc. Compumag Conf., Chicago, 1982.
 [15]
 B. H. McDonald & W. Wexler, "Mutually constrained partial differential and integral equation field formulation," in Finite Elements in Electrical and Magnetic Field Problems (M. V. U. Chari and P. Silvester, eds.), Wiley, New York, 1978.
 [16]
 I. D. Mayergoyz, Iterational Methods for the Computation of the Static Fields in Nonhomogeneous, Anisotropic and Nonlinear Medias, Naukova Dumka, Kiev, 1979. (Russian) MR 533743 (80d:78021)
 [17]
 I. D. Mayergoyz, "On numerical investigation of magnetostatic field in nonlinear ferromagnetic media," Sbornik Kibernetika, Vychislitel'naya Tekhnika, v. 17, 1972. (Russian)
 [18]
 J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.
 [19]
 J. C. Nedelec, Mixed Finite Elements in , Rapport Interne no. 49, Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau, 1979.
 [20]
 J. Pasciak, "An iterative algorithm for the volume integral method for magnetostatics problems," Comput. Math. Appl., v. 8, 1982, pp. 283290. MR 679401 (83k:78012)
 [21]
 J. Pasciak, "A new scalar potential formulation of the magnetostatic field problem," Math. Comp., v. 43, 1984, pp. 433445. MR 758192 (86f:78009)
 [22]
 I. I. Pekker, "Calculation of magnetic systems by integration over field sources," Izv. Vyssh. Uchebn. Zaved. Electromekh., v. 9, 1964, pp. 10471051 (Russian)
 [23]
 W. Rudin, Principles of Mathematical Analysis, McGrawHill, New York, 1976. MR 0385023 (52:5893)
 [24]
 J. Simkin & C. W. Trowbridge, "On the use of total scalar potential in the numerical solution of field problems in electromagnetics," Internat. J. Numer. Methods Engrg., v. 14, 1979, pp. 423440.
 [25]
 J. Simkin & C. W. Trowbridge, ThreeDimensional Computer Program (TOSCA) for Nonlinear Electromagnetic Fields, Rutherford Laboratory Report No. RL79097.
 [26]
 R. Temam, On the Theory and Numerical Analysis of the Navier Stokes Equations, NorthHolland, Amsterdam, 1977.
 [27]
 M. M. Vaĭnberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Halsted Press, New YorkToronto, 1973.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
78A30,
65N30,
7808
Retrieve articles in all journals
with MSC:
78A30,
65N30,
7808
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407581918
PII:
S 00255718(1984)07581918
Article copyright:
© Copyright 1984
American Mathematical Society
