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 = 33814

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 = 0.004852181316717192
CG iteration 2, residual = 0.002702032087071082
CG iteration 3, residual = 0.0030377840808697874
CG iteration 4, residual = 0.002255823175030216
CG iteration 5, residual = 0.0018097230540169086
CG iteration 6, residual = 0.0014942006437551627
CG iteration 7, residual = 0.0012388268033693302
CG iteration 8, residual = 0.0010294446699994242
CG iteration 9, residual = 0.0009516097931483307
CG iteration 10, residual = 0.0006604191137835776
CG iteration 11, residual = 0.0005949200300615058
CG iteration 12, residual = 0.0005054392693226594
CG iteration 13, residual = 0.0003888698384542991
CG iteration 14, residual = 0.00033806588099851543
CG iteration 15, residual = 0.00024100247060879836
CG iteration 16, residual = 0.0001835267196024368
CG iteration 17, residual = 0.00014395488754058165
CG iteration 18, residual = 0.00010695174779066355
CG iteration 19, residual = 8.938151156459134e-05
CG iteration 20, residual = 6.725501041069655e-05
CG iteration 21, residual = 4.5901465148851626e-05
CG iteration 22, residual = 3.464931310121152e-05
CG iteration 23, residual = 2.935829636411853e-05
CG iteration 24, residual = 1.9625013835855112e-05
CG iteration 25, residual = 1.6515380877046116e-05
CG iteration 26, residual = 1.1909195395244078e-05
CG iteration 27, residual = 8.96922296326759e-06
CG iteration 28, residual = 8.076287620497506e-06
CG iteration 29, residual = 5.563109692867557e-06
CG iteration 30, residual = 3.843361549385983e-06
CG iteration 31, residual = 2.8067256639464934e-06
CG iteration 32, residual = 2.2394123068129967e-06
CG iteration 33, residual = 1.591497222634232e-06
CG iteration 34, residual = 1.1931280519424174e-06
CG iteration 35, residual = 8.910716152464455e-07
CG iteration 36, residual = 6.636729734705128e-07
CG iteration 37, residual = 4.874034507469188e-07
CG iteration 38, residual = 3.63463129603561e-07
CG iteration 39, residual = 3.6518117834162186e-07
CG iteration 40, residual = 2.692473344469141e-07
CG iteration 41, residual = 1.9036940434875307e-07
CG iteration 42, residual = 1.3940681435051712e-07
CG iteration 43, residual = 1.0290704965945311e-07
CG iteration 44, residual = 1.0741791187800857e-07
CG iteration 45, residual = 7.465726692074406e-08
CG iteration 46, residual = 5.125961052586912e-08
CG iteration 47, residual = 4.699113414630372e-08
CG iteration 48, residual = 3.692042502729221e-08
CG iteration 49, residual = 2.3698980555418985e-08
CG iteration 50, residual = 1.702254735655861e-08
CG iteration 51, residual = 1.2348165943483506e-08
CG iteration 52, residual = 9.707131954766313e-09
CG iteration 53, residual = 7.521837238860094e-09
CG iteration 54, residual = 7.039456724333531e-09
CG iteration 55, residual = 4.658164612083751e-09
CG iteration 56, residual = 3.2965335675600178e-09
CG iteration 57, residual = 2.3850200103253433e-09
CG iteration 58, residual = 1.7658875768275311e-09
CG iteration 59, residual = 1.3076728296030851e-09
CG iteration 60, residual = 9.544609672880168e-10
CG iteration 61, residual = 7.290783247524824e-10
CG iteration 62, residual = 5.076933071480435e-10
CG iteration 63, residual = 4.0302177088512073e-10
CG iteration 64, residual = 3.0231235344050275e-10
CG iteration 65, residual = 2.861783404775113e-10
CG iteration 66, residual = 1.759091752336255e-10
CG iteration 67, residual = 1.199763946644875e-10
CG iteration 68, residual = 8.80026929751582e-11
CG iteration 69, residual = 9.608975863012876e-11
CG iteration 70, residual = 6.398543018634165e-11
CG iteration 71, residual = 5.5123002108111544e-11
CG iteration 72, residual = 4.03836353736656e-11
CG iteration 73, residual = 3.159283669797848e-11
CG iteration 74, residual = 2.970513075263647e-11
CG iteration 75, residual = 1.9889079212957907e-11
CG iteration 76, residual = 1.3793773523649849e-11
CG iteration 77, residual = 9.207842177951218e-12
CG iteration 78, residual = 7.665709624581502e-12
CG iteration 79, residual = 7.929621660994852e-12
CG iteration 80, residual = 4.5016021295722995e-12
CG iteration 81, residual = 3.4846402171732685e-12
CG iteration 82, residual = 3.2340228923161634e-12
CG iteration 83, residual = 2.1083402181100213e-12
CG iteration 84, residual = 1.5638446016057851e-12
CG iteration 85, residual = 1.2430235293525092e-12
CG iteration 86, residual = 9.780009818675104e-13
CG iteration 87, residual = 6.45735173702661e-13
CG iteration 88, residual = 4.793707553287705e-13
CG iteration 89, residual = 3.756960132345996e-13
CG iteration 90, residual = 2.6847407198559264e-13
CG iteration 91, residual = 2.534938474884733e-13
CG iteration 92, residual = 1.8244433401125006e-13
CG iteration 93, residual = 1.5234449561095827e-13
CG iteration 94, residual = 9.783898324517087e-14
CG iteration 95, residual = 6.783753010521209e-14
CG iteration 96, residual = 4.830336158348126e-14
CG iteration 97, residual = 3.418171996831327e-14
CG iteration 98, residual = 2.766534080791422e-14
CG iteration 99, residual = 3.052600481582997e-14
CG iteration 100, residual = 1.6342631742904282e-14
WARNING: CG did not converge to TOL

[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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