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 = 33152

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 = 0.004846766222292574
CG iteration 2, residual = 0.004063969731501546
CG iteration 3, residual = 0.0032905813897480553
CG iteration 4, residual = 0.0022607529739440317
CG iteration 5, residual = 0.001975789805600885
CG iteration 6, residual = 0.0011507895291099704
CG iteration 7, residual = 0.0008639313194094783
CG iteration 8, residual = 0.0006327449629420559
CG iteration 9, residual = 0.0004949733354845964
CG iteration 10, residual = 0.00040760541241432395
CG iteration 11, residual = 0.00031648668265075715
CG iteration 12, residual = 0.00018393955806006074
CG iteration 13, residual = 0.00013713589121295473
CG iteration 14, residual = 8.753130614937244e-05
CG iteration 15, residual = 6.404693881168134e-05
CG iteration 16, residual = 4.235156697839633e-05
CG iteration 17, residual = 2.6160932063154485e-05
CG iteration 18, residual = 1.954609606750561e-05
CG iteration 19, residual = 1.3551866855452013e-05
CG iteration 20, residual = 1.8011100336849053e-05
CG iteration 21, residual = 8.042720179646831e-06
CG iteration 22, residual = 5.687517114084214e-06
CG iteration 23, residual = 3.828297606999263e-06
CG iteration 24, residual = 2.512607985534501e-06
CG iteration 25, residual = 1.9133034156057926e-06
CG iteration 26, residual = 1.188280736862998e-06
CG iteration 27, residual = 8.414430241906949e-07
CG iteration 28, residual = 5.733998039095428e-07
CG iteration 29, residual = 3.856310050631316e-07
CG iteration 30, residual = 2.812812090503613e-07
CG iteration 31, residual = 1.789437556430661e-07
CG iteration 32, residual = 1.32013696912591e-07
CG iteration 33, residual = 1.1165487788711593e-07
CG iteration 34, residual = 8.002569549705637e-08
CG iteration 35, residual = 4.9728480104895575e-08
CG iteration 36, residual = 4.9108251591211345e-08
CG iteration 37, residual = 3.938309157648895e-08
CG iteration 38, residual = 2.169987156384291e-08
CG iteration 39, residual = 1.4457930496175452e-08
CG iteration 40, residual = 9.46664441433224e-09
CG iteration 41, residual = 6.449140628723065e-09
CG iteration 42, residual = 5.954973903218098e-09
CG iteration 43, residual = 3.929229731370565e-09
CG iteration 44, residual = 2.634847993459839e-09
CG iteration 45, residual = 1.7234152838620582e-09
CG iteration 46, residual = 1.1299005867495e-09
CG iteration 47, residual = 7.954174915607944e-10
CG iteration 48, residual = 5.005044108081919e-10
CG iteration 49, residual = 3.505163217202642e-10
CG iteration 50, residual = 2.2706320791762158e-10
CG iteration 51, residual = 1.5565324906986622e-10
CG iteration 52, residual = 1.688293110480125e-10
CG iteration 53, residual = 1.0613024745017704e-10
CG iteration 54, residual = 6.051925107315423e-11
CG iteration 55, residual = 4.0317350042260035e-11
CG iteration 56, residual = 2.7648905653190173e-11
CG iteration 57, residual = 1.786758927211454e-11
CG iteration 58, residual = 1.208458852262128e-11
CG iteration 59, residual = 7.855786379278145e-12

CG iteration 60, residual = 5.548037371290322e-12
CG iteration 61, residual = 5.582274355560327e-12
CG iteration 62, residual = 3.0885898460339927e-12
CG iteration 63, residual = 1.9778380318658677e-12
CG iteration 64, residual = 1.273016834023796e-12
CG iteration 65, residual = 8.764870229505375e-13
CG iteration 66, residual = 6.031577915487388e-13
CG iteration 67, residual = 4.685059001060606e-13
CG iteration 68, residual = 4.687208290349446e-13
CG iteration 69, residual = 2.4187141710670915e-13
CG iteration 70, residual = 2.1513991127839432e-13
CG iteration 71, residual = 1.5414636158329463e-13
CG iteration 72, residual = 8.959712425375577e-14
CG iteration 73, residual = 5.863873398499346e-14
CG iteration 74, residual = 4.024699394713719e-14
CG iteration 75, residual = 2.6754021670869943e-14
CG iteration 76, residual = 1.772276538616222e-14
CG iteration 77, residual = 1.1619608562052037e-14
CG iteration 78, residual = 7.602492448145964e-15
CG iteration 79, residual = 4.828927574807858e-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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