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 = 0.003332805101451923
CG iteration 3, residual = 0.0033139002762755526
CG iteration 4, residual = 0.0027573252392739125
CG iteration 5, residual = 0.0014699866892944622
CG iteration 6, residual = 0.0012195073011617606
CG iteration 7, residual = 0.000810408424589525
CG iteration 8, residual = 0.0006520529125240914
CG iteration 9, residual = 0.0004666815820542926
CG iteration 10, residual = 0.00035507600340398425
CG iteration 11, residual = 0.0002571940102212774
CG iteration 12, residual = 0.0001628620331813406
CG iteration 13, residual = 0.00011424246776661272
CG iteration 14, residual = 8.973472770214093e-05
CG iteration 15, residual = 5.416228079102325e-05
CG iteration 16, residual = 3.924586191413658e-05
CG iteration 17, residual = 2.7175216367202417e-05
CG iteration 18, residual = 1.8130467032714318e-05
CG iteration 19, residual = 1.3628407853491635e-05
CG iteration 20, residual = 1.0421725083585412e-05
CG iteration 21, residual = 1.2390034813806358e-05
CG iteration 22, residual = 5.513425149264709e-06
CG iteration 23, residual = 3.5878893030338235e-06
CG iteration 24, residual = 2.613374194640116e-06
CG iteration 25, residual = 1.987878963078393e-06
CG iteration 26, residual = 1.267632479036177e-06
CG iteration 27, residual = 8.694609350021191e-07
CG iteration 28, residual = 6.013535442844141e-07
CG iteration 29, residual = 4.067634925341943e-07
CG iteration 30, residual = 2.9509352359355733e-07
CG iteration 31, residual = 2.117872395231376e-07
CG iteration 32, residual = 1.3216362912051886e-07
CG iteration 33, residual = 9.609369518717728e-08
CG iteration 34, residual = 6.597524778528983e-08
CG iteration 35, residual = 4.71200107812538e-08
CG iteration 36, residual = 2.9633043010966815e-08
CG iteration 37, residual = 3.773944941124401e-08
CG iteration 38, residual = 1.9339801131688066e-08
CG iteration 39, residual = 1.3059500995641614e-08
CG iteration 40, residual = 8.520298986910211e-09
CG iteration 41, residual = 5.591493476136501e-09
CG iteration 42, residual = 3.808185401709462e-09
CG iteration 43, residual = 2.673626060562753e-09
CG iteration 44, residual = 1.871588316424271e-09
CG iteration 45, residual = 1.2905890752525892e-09
CG iteration 46, residual = 8.426050353188224e-10
CG iteration 47, residual = 5.391830050435866e-10
CG iteration 48, residual = 4.203601389060261e-10
CG iteration 49, residual = 2.81073938773017e-10
CG iteration 50, residual = 2.0086481836537293e-10
CG iteration 51, residual = 1.2659307433186293e-10
CG iteration 52, residual = 8.857816046042065e-11
CG iteration 53, residual = 1.1518417532296562e-10
CG iteration 54, residual = 5.595205346565973e-11
CG iteration 55, residual = 3.492623518431549e-11
CG iteration 56, residual = 2.435527259963421e-11
CG iteration 57, residual = 1.9590099075183922e-11
CG iteration 58, residual = 1.3428605185931181e-11
CG iteration 59, residual = 8.502412189751052e-12
CG iteration 60, residual = 5.875076989902896e-12
CG iteration 61, residual = 3.993583666411515e-12
CG iteration 62, residual = 2.6695243526301223e-12
CG iteration 63, residual = 1.7455339108042514e-12
CG iteration 64, residual = 1.2238725493729615e-12
CG iteration 65, residual = 7.550562446231048e-13
CG iteration 66, residual = 4.924890827794107e-13
CG iteration 67, residual = 3.2623805730585417e-13
CG iteration 68, residual = 2.638859376657143e-13
CG iteration 69, residual = 2.726043850363857e-13
CG iteration 70, residual = 1.2857222526010897e-13
CG iteration 71, residual = 8.643520084465284e-14
CG iteration 72, residual = 5.602466456032268e-14
CG iteration 73, residual = 3.633981507533769e-14
CG iteration 74, residual = 2.4901772913138273e-14
CG iteration 75, residual = 1.9320448536400214e-14
CG iteration 76, residual = 1.3512648281932172e-14
CG iteration 77, residual = 8.943514947258973e-15
CG iteration 78, residual = 6.341089505703736e-15
CG iteration 79, residual = 4.094320798237705e-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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