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 = 0.004809678530125143
CG iteration 2, residual = 0.0033223190132442523
CG iteration 3, residual = 0.0033115883177540243
CG iteration 4, residual = 0.0027467579779245866
CG iteration 5, residual = 0.0014658765270595371
CG iteration 6, residual = 0.001216702499924434
CG iteration 7, residual = 0.0008096582309573287
CG iteration 8, residual = 0.0006570076838868013
CG iteration 9, residual = 0.00047508691834428814
CG iteration 10, residual = 0.00036224214074476
CG iteration 11, residual = 0.0002544183424853318
CG iteration 12, residual = 0.0001619498909443696
CG iteration 13, residual = 0.00011358302702052086
CG iteration 14, residual = 8.950382818008153e-05
CG iteration 15, residual = 5.3941268122952725e-05
CG iteration 16, residual = 3.9160417882599236e-05
CG iteration 17, residual = 2.708231819649529e-05
CG iteration 18, residual = 1.8249643282336552e-05
CG iteration 19, residual = 1.358736605172267e-05
CG iteration 20, residual = 9.94176247212281e-06
CG iteration 21, residual = 1.2024195323061934e-05
CG iteration 22, residual = 5.647191374211599e-06
CG iteration 23, residual = 3.59364635310983e-06
CG iteration 24, residual = 2.5589068701358585e-06
CG iteration 25, residual = 1.977947764187858e-06
CG iteration 26, residual = 1.2770934676740194e-06
CG iteration 27, residual = 8.740832403623361e-07
CG iteration 28, residual = 5.927228746857172e-07
CG iteration 29, residual = 4.0089017034467943e-07
CG iteration 30, residual = 2.8564581969581914e-07
CG iteration 31, residual = 2.0292685846580763e-07
CG iteration 32, residual = 1.319963698675324e-07
CG iteration 33, residual = 9.850315519015888e-08
CG iteration 34, residual = 6.734296682814417e-08
CG iteration 35, residual = 4.756021419920872e-08
CG iteration 36, residual = 2.9180212579662152e-08
CG iteration 37, residual = 2.5955030872539008e-08
CG iteration 38, residual = 2.3679911979105457e-08
CG iteration 39, residual = 1.304058953486766e-08
CG iteration 40, residual = 8.395592563601195e-09
CG iteration 41, residual = 5.711979790332258e-09
CG iteration 42, residual = 4.1810498498190084e-09
CG iteration 43, residual = 2.794949339588557e-09
CG iteration 44, residual = 1.8595331198167794e-09
CG iteration 45, residual = 1.2654495208227433e-09
CG iteration 46, residual = 8.583836413660985e-10
CG iteration 47, residual = 5.766587486128901e-10
CG iteration 48, residual = 4.217770573065599e-10
CG iteration 49, residual = 2.7611892452267563e-10
CG iteration 50, residual = 2.1705324158494076e-10
CG iteration 51, residual = 1.2859604144828741e-10
CG iteration 52, residual = 8.652630936907714e-11
CG iteration 53, residual = 7.529522122778294e-11
CG iteration 54, residual = 6.973696391657553e-11
CG iteration 55, residual = 3.513889127070108e-11
CG iteration 56, residual = 2.6106328733146427e-11
CG iteration 57, residual = 2.0977798794771095e-11
CG iteration 58, residual = 1.2790486410121312e-11
CG iteration 59, residual = 8.489642990779244e-12
CG iteration 60, residual = 5.922675768750128e-12
CG iteration 61, residual = 3.942553777764441e-12
CG iteration 62, residual = 2.695611299715122e-12
CG iteration 63, residual = 1.7387075286535152e-12
CG iteration 64, residual = 1.2104136890035807e-12
CG iteration 65, residual = 7.623361075745722e-13
CG iteration 66, residual = 4.941737042305336e-13
CG iteration 67, residual = 3.208141848528967e-13
CG iteration 68, residual = 2.2108033738514836e-13
CG iteration 69, residual = 3.03947866252762e-13
CG iteration 70, residual = 1.3744434355480244e-13
CG iteration 71, residual = 9.424145929857764e-14
CG iteration 72, residual = 6.131163789688446e-14
CG iteration 73, residual = 4.206920018189592e-14
CG iteration 74, residual = 2.7626192747036357e-14
CG iteration 75, residual = 1.7654971503490337e-14
CG iteration 76, residual = 1.1562456282932595e-14
CG iteration 77, residual = 8.186207544335052e-15
CG iteration 78, residual = 6.5138679350705406e-15
CG iteration 79, residual = 4.145718084921741e-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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