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 = 33832

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 = 0.0048217200778921
CG iteration 2, residual = 0.0032104516402596908
CG iteration 3, residual = 0.002371542910649177
CG iteration 4, residual = 0.00163220430908912
CG iteration 5, residual = 0.0012442854977395656
CG iteration 6, residual = 0.0009044032351893669
CG iteration 7, residual = 0.0007001208867136664
CG iteration 8, residual = 0.000560853226580446
CG iteration 9, residual = 0.00036387905804978536
CG iteration 10, residual = 0.00029195248969641433
CG iteration 11, residual = 0.00018574649944941198
CG iteration 12, residual = 0.00013580261400185465
CG iteration 13, residual = 9.190407945869495e-05
CG iteration 14, residual = 5.7868673030200645e-05
CG iteration 15, residual = 4.41336643641394e-05
CG iteration 16, residual = 3.1444318775645386e-05
CG iteration 17, residual = 1.9528671630726745e-05
CG iteration 18, residual = 1.3517471299616875e-05
CG iteration 19, residual = 9.890880398831075e-06
CG iteration 20, residual = 7.153565906016244e-06
CG iteration 21, residual = 4.4474035611053765e-06
CG iteration 22, residual = 3.137365461497175e-06
CG iteration 23, residual = 2.128992097134961e-06
CG iteration 24, residual = 1.538291064437111e-06
CG iteration 25, residual = 1.0783724452556604e-06
CG iteration 26, residual = 7.606999859263029e-07
CG iteration 27, residual = 5.148982458847621e-07
CG iteration 28, residual = 3.3600038315788276e-07
CG iteration 29, residual = 2.3316587903829399e-07
CG iteration 30, residual = 1.624295539094728e-07
CG iteration 31, residual = 1.0751611046927512e-07
CG iteration 32, residual = 7.593395585516964e-08
CG iteration 33, residual = 5.2038573458481385e-08
CG iteration 34, residual = 3.5964934755633967e-08
CG iteration 35, residual = 2.30817306643575e-08
CG iteration 36, residual = 1.5146018252582007e-08
CG iteration 37, residual = 1.0485752616422107e-08
CG iteration 38, residual = 7.486630162118141e-09
CG iteration 39, residual = 4.871917410605163e-09
CG iteration 40, residual = 3.1569591268319432e-09
CG iteration 41, residual = 2.0978635926047406e-09
CG iteration 42, residual = 1.4148898470566736e-09
CG iteration 43, residual = 1.016043978848877e-09
CG iteration 44, residual = 6.42701387018596e-10
CG iteration 45, residual = 4.4248346711836244e-10
CG iteration 46, residual = 2.9158545508656925e-10
CG iteration 47, residual = 1.9285756640215358e-10
CG iteration 48, residual = 1.3772381746656972e-10
CG iteration 49, residual = 8.94163520569618e-11
CG iteration 50, residual = 6.023956841762242e-11
CG iteration 51, residual = 4.927397010689627e-11
CG iteration 52, residual = 3.36355810002151e-11
CG iteration 53, residual = 2.2026374107887504e-11
CG iteration 54, residual = 1.4180255085540377e-11
CG iteration 55, residual = 9.467053451258482e-12
CG iteration 56, residual = 6.3440009010984225e-12
CG iteration 57, residual = 4.116201946573997e-12
CG iteration 58, residual = 2.7335531041648592e-12
CG iteration 59, residual = 1.8638530252858742e-12
CG iteration 60, residual = 1.2738964466600408e-12
CG iteration 61, residual = 8.306915786511626e-13
CG iteration 62, residual = 5.761367067852234e-13
CG iteration 63, residual = 3.9536103854836927e-13
CG iteration 64, residual = 2.484851362220686e-13
CG iteration 65, residual = 1.6541942517897304e-13
CG iteration 66, residual = 1.14979988830305e-13
CG iteration 67, residual = 7.664317738266347e-14
CG iteration 68, residual = 5.92489266355865e-14
CG iteration 69, residual = 4.010911251451925e-14
CG iteration 70, residual = 2.7480568686555426e-14
CG iteration 71, residual = 1.7692514456867963e-14
CG iteration 72, residual = 1.2694706063132632e-14
CG iteration 73, residual = 9.749032883625256e-15
CG iteration 74, residual = 6.283299330225406e-15
CG iteration 75, residual = 4.207822188487633e-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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