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 = 0.004814181258911799
CG iteration 2, residual = 0.003332805101451917
CG iteration 3, residual = 0.003313900276275579
CG iteration 4, residual = 0.002757325239273891
CG iteration 5, residual = 0.0014699866892944596
CG iteration 6, residual = 0.0012195073011617577
CG iteration 7, residual = 0.0008104084245895238
CG iteration 8, residual = 0.0006520529125240902
CG iteration 9, residual = 0.0004666815820542918
CG iteration 10, residual = 0.0003550760034039838
CG iteration 11, residual = 0.00025719401022127693
CG iteration 12, residual = 0.00016286203318134043
CG iteration 13, residual = 0.00011424246776661252
CG iteration 14, residual = 8.973472770213764e-05
CG iteration 15, residual = 5.4162280790893847e-05
CG iteration 16, residual = 3.924586190701206e-05
CG iteration 17, residual = 2.717521611340987e-05
CG iteration 18, residual = 1.813045471164716e-05
CG iteration 19, residual = 1.362789760363238e-05
CG iteration 20, residual = 1.0410718906485234e-05
CG iteration 21, residual = 1.2395238691642852e-05
CG iteration 22, residual = 5.51461243084891e-06
CG iteration 23, residual = 3.587894972989454e-06
CG iteration 24, residual = 2.6133742559919905e-06
CG iteration 25, residual = 1.9878789643052452e-06
CG iteration 26, residual = 1.267632479050345e-06
CG iteration 27, residual = 8.694609350022428e-07
CG iteration 28, residual = 6.013535442844173e-07
CG iteration 29, residual = 4.067634925341961e-07
CG iteration 30, residual = 2.9509352359355225e-07
CG iteration 31, residual = 2.1178723952291813e-07
CG iteration 32, residual = 1.3216362910978648e-07
CG iteration 33, residual = 9.60936945386587e-08
CG iteration 34, residual = 6.597521978645465e-08
CG iteration 35, residual = 4.711870149967481e-08
CG iteration 36, residual = 2.9567647439119096e-08
CG iteration 37, residual = 3.6770434762680685e-08
CG iteration 38, residual = 1.948940864340131e-08
CG iteration 39, residual = 1.30613085438085e-08
CG iteration 40, residual = 8.520313974459386e-09
CG iteration 41, residual = 5.591493570245531e-09
CG iteration 42, residual = 3.808185402308496e-09
CG iteration 43, residual = 2.6736260605470346e-09
CG iteration 44, residual = 1.87158831635027e-09
CG iteration 45, residual = 1.2905890750677464e-09
CG iteration 46, residual = 8.426050344612989e-10
CG iteration 47, residual = 5.391829984926165e-10
CG iteration 48, residual = 4.203601084650961e-10
CG iteration 49, residual = 2.8107388828587115e-10
CG iteration 50, residual = 2.0086435269710062e-10
CG iteration 51, residual = 1.265759601570525e-10
CG iteration 52, residual = 8.73539773685684e-11
CG iteration 53, residual = 1.0870664080539523e-10
CG iteration 54, residual = 5.706973863087455e-11
CG iteration 55, residual = 3.4878173539555605e-11
CG iteration 56, residual = 2.3967996432985348e-11
CG iteration 57, residual = 1.8731795436651616e-11
CG iteration 58, residual = 1.360899007804432e-11
CG iteration 59, residual = 8.537810438713345e-12
CG iteration 60, residual = 5.878017065471493e-12
CG iteration 61, residual = 3.99386806032481e-12
CG iteration 62, residual = 2.6695508843492858e-12
CG iteration 63, residual = 1.7455353302280171e-12
CG iteration 64, residual = 1.2238726876522724e-12
CG iteration 65, residual = 7.550562170953146e-13
CG iteration 66, residual = 4.924856563738077e-13
CG iteration 67, residual = 3.2593386987211954e-13
CG iteration 68, residual = 2.4563472248528863e-13
CG iteration 69, residual = 2.9400001087057896e-13
CG iteration 70, residual = 1.2893262479345654e-13
CG iteration 71, residual = 8.643861017121089e-14
CG iteration 72, residual = 5.6032965978430566e-14
CG iteration 73, residual = 3.639592122353495e-14
CG iteration 74, residual = 2.5261715611931687e-14
CG iteration 75, residual = 1.981645481419682e-14
CG iteration 76, residual = 1.3508129013076084e-14
CG iteration 77, residual = 8.916149033803626e-15
CG iteration 78, residual = 6.336428072720218e-15
CG iteration 79, residual = 4.095042625233966e-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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