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 = 0.004814181258911797
CG iteration 2, residual = 0.003332805101451917
CG iteration 3, residual = 0.0033139002762755526
CG iteration 4, residual = 0.002757325239273908
CG iteration 5, residual = 0.00146998668929446
CG iteration 6, residual = 0.0012195073011617582
CG iteration 7, residual = 0.0008104084245895247
CG iteration 8, residual = 0.0006520529125240901
CG iteration 9, residual = 0.00046668158205429277
CG iteration 10, residual = 0.00035507600340398414
CG iteration 11, residual = 0.0002571940102212772
CG iteration 12, residual = 0.00016286203318134046
CG iteration 13, residual = 0.00011424246776661388
CG iteration 14, residual = 8.97347277021816e-05
CG iteration 15, residual = 5.416228079257075e-05
CG iteration 16, residual = 3.924586199928558e-05
CG iteration 17, residual = 2.7175219400455062e-05
CG iteration 18, residual = 1.813061428933163e-05
CG iteration 19, residual = 1.3634503200727514e-05
CG iteration 20, residual = 1.0550947558511082e-05
CG iteration 21, residual = 1.2312498050231892e-05
CG iteration 22, residual = 5.501385206290639e-06
CG iteration 23, residual = 3.5878319659771296e-06
CG iteration 24, residual = 2.61337357423878e-06
CG iteration 25, residual = 1.987878950672272e-06
CG iteration 26, residual = 1.2676324788929045e-06
CG iteration 27, residual = 8.694609350008714e-07
CG iteration 28, residual = 6.013535442843971e-07
CG iteration 29, residual = 4.0676349253418984e-07
CG iteration 30, residual = 2.9509352359356194e-07
CG iteration 31, residual = 2.1178723952337746e-07
CG iteration 32, residual = 1.3216362913237333e-07
CG iteration 33, residual = 9.609369590359063e-08
CG iteration 34, residual = 6.597527871539334e-08
CG iteration 35, residual = 4.7121457056310736e-08
CG iteration 36, residual = 2.970500212124568e-08
CG iteration 37, residual = 3.863264907210887e-08
CG iteration 38, residual = 1.9210140361688518e-08
CG iteration 39, residual = 1.305797066752995e-08
CG iteration 40, residual = 8.520286302628786e-09
CG iteration 41, residual = 5.591493396489113e-09
CG iteration 42, residual = 3.808185401193775e-09
CG iteration 43, residual = 2.673626060532629e-09
CG iteration 44, residual = 1.8715883163330725e-09
CG iteration 45, residual = 1.2905890750250972e-09
CG iteration 46, residual = 8.426050342635018e-10
CG iteration 47, residual = 5.391829969841092e-10
CG iteration 48, residual = 4.203601015986842e-10
CG iteration 49, residual = 2.8107388153477273e-10
CG iteration 50, residual = 2.0086445988503746e-10
CG iteration 51, residual = 1.2658212796558968e-10
CG iteration 52, residual = 8.781213041233486e-11
CG iteration 53, residual = 1.1186209446714407e-10
CG iteration 54, residual = 5.6527149650251617e-11
CG iteration 55, residual = 3.486154790179378e-11
CG iteration 56, residual = 2.387482556164525e-11
CG iteration 57, residual = 1.8455755430782637e-11
CG iteration 58, residual = 1.3653926082408187e-11
CG iteration 59, residual = 8.55241974465924e-12
CG iteration 60, residual = 5.879281064100611e-12
CG iteration 61, residual = 3.993991043035662e-12
CG iteration 62, residual = 2.6695623665937325e-12
CG iteration 63, residual = 1.7455359445448894e-12
CG iteration 64, residual = 1.2238727475104924e-12
CG iteration 65, residual = 7.550562108244841e-13
CG iteration 66, residual = 4.924847619948635e-13
CG iteration 67, residual = 3.258546197363455e-13
CG iteration 68, residual = 2.4045516217647757e-13
CG iteration 69, residual = 3.011563633573595e-13
CG iteration 70, residual = 1.291252109310869e-13
CG iteration 71, residual = 8.643682048933794e-14
CG iteration 72, residual = 5.601712476244286e-14
CG iteration 73, residual = 3.628833995289147e-14
CG iteration 74, residual = 2.4552392855996037e-14
CG iteration 75, residual = 1.8604174671652697e-14
CG iteration 76, residual = 1.338617339616166e-14
CG iteration 77, residual = 8.988483429314355e-15
CG iteration 78, residual = 6.352494759893901e-15
CG iteration 79, residual = 4.0969473244654025e-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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