Numerical Modeling in Magnetism
Macro-Magnetism: Solution of Maxwell´s Equations – Engineering of (electro)magnetic devices
Atomic Magnetism: Instrinsic Magnetic Properties
Micromagnetism: Domain Dynamics, Hysteresis
MFM image
Micromagnetic simulation.
Macro-MagnetismCalculate Magnetic Fields and Forces
BIOT-SAVART Law
'
'
')'(
4)( 3
30 xd
xx
xxxxB
J
Maxwells Laws
in Matter
)()(
0)(
0 xjxB
xB
)()(
)()()(
)()()(
)()(
0)(
00
00
0
xjxH
xMxBxH
xMxjxB
xjxM
jj
xB
micro
micro
J
MH
)()( xBxA
Vector Potential
...if j=0, determination ofH with „Magnetic Charges“
Boundary Element Method (BEM)The Boundary Element Method solves field problems by solving an equivalent source problem. In the case of electric fields it solves for equivalent charge, while in the case of magnetic fields it solves for equivalent currents.
BEM also uses an integral formulation of Maxwell's Equations, which allow for very highly accurate field calculations. The formulations are such that differentiating the field is not a problem.
Problem
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'
')'(
4)( 3
30 xd
xx
xxxjxB
Finite Element Method (FEM)Finite elements solve by breaking up a problem into small regions and solutions are found for each region taking into account only the regions that are right next to the one being solved. In the case of magnetic fields where FEM is often used, the vector potential is what is solved for in these regions. Magnetic field solutions are derived from the vector potential through differentiating the solution. This can cause problems in smoothness of field solutions. Theoretically, any partial differential equation class of problem can be solved using FEM (although some types will do better than others.)
Problem
)()( xjxA
•BEM Advantages •FEM Advantages
•Open regions not a problem •Extreme aspect ratios not a problem •World space is not truncated •Perfectly smooth field solutions
•Non linear problems easily solved •Easy formulations allow many different types of problems to be solved
•BEM Disadvantages •FEM Disadvantages
•Non linear problems difficult to solve •Some problem formulations are not possible
•Not well suited for open region problems •Extreme aspect ratios can cause problems •Truncated world space •Fields can be noisy
Hybrid BE-FE Method
When you compare the advantages and disadvantages (below), you can see that an ideal method would be to combine both methods to form a hybrid of the two. •The hybrid solver takes the strengths of both methods and uses them to an advantage. •The BEM handles the open regions and linear solution while the FEM handles the non-linear areas. •The results are excellent open region solutions with no non-linear convergence problems.
EXAMPLE
FORCES ON CRYOMAGNET OF PANDA
• vertical field Hmax = 15T
PANDA – general view
beam-channelmonochromator-shielding with platform
Cabin with computer work-placesand electronics
secondary spectrometerwith surrounding radioprotection
PANDA – general view
CostsCryomagnet: 700.000€100% stainless steel concrete screening: approx 3.000.000€Concrete screening with magnetic steel: 600.000€
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