This page was generated from unit-2.4-Maxwell/Maxwell.ipynb.

2.4 Maxwell’s Equations

[Peter Monk: "Finite Elements for Maxwell’s Equations"]

Magnetostatic field generated by a permanent magnet

magnetic flux \(B\), magnetic field \(H\), given magnetization \(M\):

\[\DeclareMathOperator{\Grad}{grad} \DeclareMathOperator{\Curl}{curl} \DeclareMathOperator{\Div}{div} B = \mu (H + M), \quad \Div B = 0, \quad \Curl H = 0\]

Introducing a vector-potential \(A\) such that \(B = \Curl A\), and putting equations together we get

\[\Curl \mu^{-1} \Curl A = \Curl M\]

In weak form: Find \(A \in H(\Curl)\) such that

\[\int \mu^{-1} \Curl A \Curl v = \int M \Curl v \qquad \forall \, v \in H(\Curl)\]

Usually, the permeability \(\mu\) is given as \(\mu = \mu_r \mu_0\), with \(\mu_0 = 4 \pi 10^{-7}\) the permeability of vacuum.

[1]:

from ngsolve import *
from ngsolve.webgui import Draw
from netgen.occ import *

Geometric model and meshing of a bar magnet:

[2]:

# box = OrthoBrick(Pnt(-3,-3,-3),Pnt(3,3,3)).bc("outer")
# magnet = Cylinder(Pnt(-1,0,0),Pnt(1,0,0), 0.3) * OrthoBrick(Pnt(-1,-3,-3),Pnt(1,3,3))
# air = box - magnet
box = Box( (-3,-3,-3), (3,3,3))
box.faces.name = "outer"

magnet = Cylinder((-1,0,0),X, r=0.3, h=2)
magnet.mat("magnet")
magnet.faces.col = (1,0,0)

air = box-magnet
air.mat("air")
shape = Glue([air,magnet])
geo = OCCGeometry(shape)

Draw (shape, clipping={ "z" : -1, "function":True})

mesh = Mesh(geo.GenerateMesh(maxh=2, curvaturesafety=1))
mesh.Curve(3);

[3]:

mesh.GetMaterials(), mesh.GetBoundaries()

[3]:

(('air', 'magnet'),
 ('outer',
  'outer',
  'outer',
  'outer',
  'outer',
  'outer',
  'default',
  'default',
  'default'))

Define space, forms and preconditioner.

  • To obtain a regular system matrix, we regularize by adding a very small \(L_2\) term.

  • We solve magnetostatics, so we can gauge by adding and arbitrary gradient field. A cheap possibility is to delete all basis-functions which are gradients (flag 'nograds')

[4]:

fes = HCurl(mesh, order=3, dirichlet="outer", nograds=True)
print ("ndof =", fes.ndof)
u,v = fes.TnT()

from math import pi
mu0 = 4*pi*1e-7
mur = mesh.MaterialCF({"magnet" : 1000}, default=1)

a = BilinearForm(fes)
a += 1/(mu0*mur)*curl(u)*curl(v)*dx + 1e-8/(mu0*mur)*u*v*dx
c = Preconditioner(a, "bddc")

f = LinearForm(fes)
mag = mesh.MaterialCF({"magnet" : (1,0,0)}, default=(0,0,0))
f += mag*curl(v) * dx("magnet")

ndof = 32867

Assemble system and setup preconditioner using task-parallelization:

[5]:

with TaskManager():
    a.Assemble()
    f.Assemble()

Finally, declare GridFunction and solve by preconditioned CG iteration:

[6]:

gfu = GridFunction(fes)
with TaskManager():
    solvers.CG(sol=gfu.vec, rhs=f.vec, mat=a.mat, pre=c.mat, printrates=True)

CG iteration 1, residual = 0.004809678530125144
CG iteration 2, residual = 0.003322319013244251
CG iteration 3, residual = 0.0033115883177540096
CG iteration 4, residual = 0.0027467579779245845
CG iteration 5, residual = 0.001465876527059534
CG iteration 6, residual = 0.001216702499924431
CG iteration 7, residual = 0.0008096582309573265
CG iteration 8, residual = 0.0006570076838867998
CG iteration 9, residual = 0.0004750869183442867
CG iteration 10, residual = 0.00036224214074475936
CG iteration 11, residual = 0.00025441834248533086
CG iteration 12, residual = 0.000161949890944369
CG iteration 13, residual = 0.00011358302702052068
CG iteration 14, residual = 8.950382818008931e-05
CG iteration 15, residual = 5.394126812325214e-05
CG iteration 16, residual = 3.916041789884022e-05
CG iteration 17, residual = 2.708231878177548e-05
CG iteration 18, residual = 1.8249670625000672e-05
CG iteration 19, residual = 1.358847015749282e-05
CG iteration 20, residual = 9.969167224172471e-06
CG iteration 21, residual = 1.2104265738248299e-05
CG iteration 22, residual = 5.634203888689279e-06
CG iteration 23, residual = 3.593580605201803e-06
CG iteration 24, residual = 2.5589062399881085e-06
CG iteration 25, residual = 1.977947752484014e-06
CG iteration 26, residual = 1.2770934675198277e-06
CG iteration 27, residual = 8.740832403609009e-07
CG iteration 28, residual = 5.927228746857036e-07
CG iteration 29, residual = 4.008901703446804e-07
CG iteration 30, residual = 2.856458196958198e-07
CG iteration 31, residual = 2.0292685846581686e-07
CG iteration 32, residual = 1.3199636986796917e-07
CG iteration 33, residual = 9.850315521100438e-08
CG iteration 34, residual = 6.734296756407342e-08
CG iteration 35, residual = 4.756024628807897e-08
CG iteration 36, residual = 2.918200143772605e-08
CG iteration 37, residual = 2.6048516623956254e-08
CG iteration 38, residual = 2.3618967058656036e-08
CG iteration 39, residual = 1.3038899529519846e-08
CG iteration 40, residual = 8.395580787423862e-09
CG iteration 41, residual = 5.71197971025939e-09
CG iteration 42, residual = 4.181049848958194e-09
CG iteration 43, residual = 2.7949493395722683e-09
CG iteration 44, residual = 1.8595331197798458e-09
CG iteration 45, residual = 1.265449520642666e-09
CG iteration 46, residual = 8.583836403658977e-10
CG iteration 47, residual = 5.766587434601848e-10
CG iteration 48, residual = 4.2177704045293713e-10
CG iteration 49, residual = 2.761188637245717e-10
CG iteration 50, residual = 2.1705309020956551e-10
CG iteration 51, residual = 1.2859418315228727e-10
CG iteration 52, residual = 8.638225561945725e-11
CG iteration 53, residual = 6.83997517800194e-11
CG iteration 54, residual = 7.600321432428201e-11
CG iteration 55, residual = 3.5200595805331367e-11
CG iteration 56, residual = 2.581611872925598e-11
CG iteration 57, residual = 2.0964133821773526e-11
CG iteration 58, residual = 1.2820394174962313e-11
CG iteration 59, residual = 8.491859526968079e-12
CG iteration 60, residual = 5.922889679323463e-12
CG iteration 61, residual = 3.942571748271215e-12
CG iteration 62, residual = 2.695612992758695e-12
CG iteration 63, residual = 1.7387076275367228e-12
CG iteration 64, residual = 1.2104136967605236e-12
CG iteration 65, residual = 7.623360996818055e-13
CG iteration 66, residual = 4.941729157275795e-13
CG iteration 67, residual = 3.207401109390441e-13
CG iteration 68, residual = 2.1488594582316704e-13
CG iteration 69, residual = 2.470973754503495e-13
CG iteration 70, residual = 1.467637896788926e-13
CG iteration 71, residual = 9.434247253315428e-14
CG iteration 72, residual = 6.131472781958991e-14
CG iteration 73, residual = 4.207767072420215e-14
CG iteration 74, residual = 2.7659590529935987e-14
CG iteration 75, residual = 1.7857897250793696e-14
CG iteration 76, residual = 1.2922149408279961e-14
CG iteration 77, residual = 1.0002886428129033e-14
CG iteration 78, residual = 6.507402892226719e-15
CG iteration 79, residual = 3.974071670618134e-15

[7]:

Draw (curl(gfu), mesh, "B-field", draw_surf=False, \
      clipping = { "z" : -1, "function":True}, \
      vectors = { "grid_size":50}, min=0, max=2e-5);

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