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

'

'

')'(

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€