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 = 0.004814181258911796
CG iteration 2, residual = 0.0033328051014519266
CG iteration 3, residual = 0.003313900276275559
CG iteration 4, residual = 0.0027573252392739172
CG iteration 5, residual = 0.0014699866892944626
CG iteration 6, residual = 0.0012195073011617612
CG iteration 7, residual = 0.0008104084245895261
CG iteration 8, residual = 0.000652052912524092
CG iteration 9, residual = 0.0004666815820542941
CG iteration 10, residual = 0.000355076003403985
CG iteration 11, residual = 0.00025719401022127763
CG iteration 12, residual = 0.00016286203318134092
CG iteration 13, residual = 0.00011424246776661325
CG iteration 14, residual = 8.973472770214863e-05
CG iteration 15, residual = 5.4162280791303336e-05
CG iteration 16, residual = 3.924586192954367e-05
CG iteration 17, residual = 2.717521691605227e-05
CG iteration 18, residual = 1.813049367811938e-05
CG iteration 19, residual = 1.3629511183362454e-05
CG iteration 20, residual = 1.044542299686822e-05
CG iteration 21, residual = 1.2377967926798301e-05
CG iteration 22, residual = 5.510964828249584e-06
CG iteration 23, residual = 3.587877562647584e-06
CG iteration 24, residual = 2.6133740676039056e-06
CG iteration 25, residual = 1.9878789605380586e-06
CG iteration 26, residual = 1.2676324790068397e-06
CG iteration 27, residual = 8.694609350018629e-07
CG iteration 28, residual = 6.013535442844106e-07
CG iteration 29, residual = 4.067634925341926e-07
CG iteration 30, residual = 2.950935235935484e-07
CG iteration 31, residual = 2.1178723952289645e-07
CG iteration 32, residual = 1.3216362910875341e-07
CG iteration 33, residual = 9.60936944762761e-08
CG iteration 34, residual = 6.597521709322192e-08
CG iteration 35, residual = 4.711857555518826e-08
CG iteration 36, residual = 2.956134408394797e-08
CG iteration 37, residual = 3.666790630798605e-08
CG iteration 38, residual = 1.9505883874299684e-08
CG iteration 39, residual = 1.306151040586175e-08
CG iteration 40, residual = 8.52031564859815e-09
CG iteration 41, residual = 5.5914935807589844e-09
CG iteration 42, residual = 3.808185402384811e-09
CG iteration 43, residual = 2.673626060592406e-09
CG iteration 44, residual = 1.8715883165091217e-09
CG iteration 45, residual = 1.2905890754642937e-09
CG iteration 46, residual = 8.426050363009281e-10
CG iteration 47, residual = 5.391830125370951e-10
CG iteration 48, residual = 4.203601732000145e-10
CG iteration 49, residual = 2.8107397860719865e-10
CG iteration 50, residual = 2.0086456295009973e-10
CG iteration 51, residual = 1.2657549853158903e-10
CG iteration 52, residual = 8.726957735435648e-11
CG iteration 53, residual = 1.0793950756002208e-10
CG iteration 54, residual = 5.7223207749085994e-11
CG iteration 55, residual = 3.4998314852663266e-11
CG iteration 56, residual = 2.4768577364041837e-11
CG iteration 57, residual = 2.015479266947901e-11
CG iteration 58, residual = 1.3283243386662952e-11
CG iteration 59, residual = 8.48398301009249e-12
CG iteration 60, residual = 5.8736099415130256e-12
CG iteration 61, residual = 3.993442620027092e-12
CG iteration 62, residual = 2.6695112047852878e-12
CG iteration 63, residual = 1.7455332074433274e-12
CG iteration 64, residual = 1.223872480118271e-12
CG iteration 65, residual = 7.550562043967364e-13
CG iteration 66, residual = 4.92485482715555e-13
CG iteration 67, residual = 3.259186966215252e-13
CG iteration 68, residual = 2.446588634631383e-13
CG iteration 69, residual = 2.9532460642892923e-13
CG iteration 70, residual = 1.2896353467598483e-13
CG iteration 71, residual = 8.643571512537927e-14
CG iteration 72, residual = 5.601578621323379e-14
CG iteration 73, residual = 3.627914777904987e-14
CG iteration 74, residual = 2.4487481709241304e-14
CG iteration 75, residual = 1.8434560147937982e-14
CG iteration 76, residual = 1.331876744387435e-14
CG iteration 77, residual = 8.99764544948732e-15
CG iteration 78, residual = 6.355839811501196e-15
CG iteration 79, residual = 4.099025764718431e-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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