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 = 0.0048096785301251456
CG iteration 2, residual = 0.0033223190132442367
CG iteration 3, residual = 0.0033115883177540035
CG iteration 4, residual = 0.0027467579779245745
CG iteration 5, residual = 0.0014658765270595291
CG iteration 6, residual = 0.001216702499924429
CG iteration 7, residual = 0.0008096582309573252
CG iteration 8, residual = 0.0006570076838867986
CG iteration 9, residual = 0.00047508691834428495
CG iteration 10, residual = 0.00036224214074475784
CG iteration 11, residual = 0.0002544183424853304
CG iteration 12, residual = 0.00016194989094436873
CG iteration 13, residual = 0.00011358302702051966
CG iteration 14, residual = 8.950382818007012e-05
CG iteration 15, residual = 5.394126812253756e-05
CG iteration 16, residual = 3.916041786010337e-05
CG iteration 17, residual = 2.7082317385818408e-05
CG iteration 18, residual = 1.8249605409666397e-05
CG iteration 19, residual = 1.3585836443955678e-05
CG iteration 20, residual = 9.903505582983726e-06
CG iteration 21, residual = 1.1887349542914524e-05
CG iteration 22, residual = 5.668463550039483e-06
CG iteration 23, residual = 3.593754842733219e-06
CG iteration 24, residual = 2.558907910002654e-06
CG iteration 25, residual = 1.9779477835014837e-06
CG iteration 26, residual = 1.2770934679284558e-06
CG iteration 27, residual = 8.740832403646971e-07
CG iteration 28, residual = 5.927228746857349e-07
CG iteration 29, residual = 4.00890170344678e-07
CG iteration 30, residual = 2.856458196958168e-07
CG iteration 31, residual = 2.0292685846572838e-07
CG iteration 32, residual = 1.3199636986386974e-07
CG iteration 33, residual = 9.850315501541405e-08
CG iteration 34, residual = 6.734296065880552e-08
CG iteration 35, residual = 4.7559945194692385e-08
CG iteration 36, residual = 2.9165209468135973e-08
CG iteration 37, residual = 2.514110554512016e-08
CG iteration 38, residual = 2.4248388624833317e-08
CG iteration 39, residual = 1.3058168493959485e-08
CG iteration 40, residual = 8.39571530241003e-09
CG iteration 41, residual = 5.711980624907463e-09
CG iteration 42, residual = 4.18104985874396e-09
CG iteration 43, residual = 2.7949493396367437e-09
CG iteration 44, residual = 1.8595331195517175e-09
CG iteration 45, residual = 1.2654495195245338e-09
CG iteration 46, residual = 8.583836341553482e-10
CG iteration 47, residual = 5.76658711468333e-10
CG iteration 48, residual = 4.2177693595999993e-10
CG iteration 49, residual = 2.7611849396547544e-10
CG iteration 50, residual = 2.1705243172761837e-10
CG iteration 51, residual = 1.285935097330482e-10
CG iteration 52, residual = 8.642552233514406e-11
CG iteration 53, residual = 7.119380617628866e-11
CG iteration 54, residual = 7.307379233550981e-11
CG iteration 55, residual = 3.482727245850231e-11
CG iteration 56, residual = 2.356359180304191e-11
CG iteration 57, residual = 1.8679195822137377e-11
CG iteration 58, residual = 1.3430458388623504e-11
CG iteration 59, residual = 8.571090540201524e-12
CG iteration 60, residual = 5.931149381998368e-12
CG iteration 61, residual = 3.94327317177945e-12
CG iteration 62, residual = 2.6956791637327447e-12
CG iteration 63, residual = 1.73871149259715e-12
CG iteration 64, residual = 1.2104140033040437e-12
CG iteration 65, residual = 7.623361221004217e-13
CG iteration 66, residual = 4.941734125999547e-13
CG iteration 67, residual = 3.207867489185742e-13
CG iteration 68, residual = 2.1881813094540718e-13
CG iteration 69, residual = 2.948593701775262e-13
CG iteration 70, residual = 1.3885538700072172e-13
CG iteration 71, residual = 9.425547027472879e-14
CG iteration 72, residual = 6.131210157646456e-14
CG iteration 73, residual = 4.2070494800309136e-14
CG iteration 74, residual = 2.763128377363373e-14
CG iteration 75, residual = 1.768633280137153e-14
CG iteration 76, residual = 1.181259062617842e-14
CG iteration 77, residual = 9.044372718951105e-15
CG iteration 78, residual = 6.755786464444786e-15
CG iteration 79, residual = 4.020192768300405e-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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