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 = 0.004821720077892098
CG iteration 2, residual = 0.0032104516402596942
CG iteration 3, residual = 0.0023715429106491775
CG iteration 4, residual = 0.001632204309089121
CG iteration 5, residual = 0.0012442854977395665
CG iteration 6, residual = 0.0009044032351893677
CG iteration 7, residual = 0.0007001208867136678
CG iteration 8, residual = 0.0005608532265804472
CG iteration 9, residual = 0.00036387905804978585
CG iteration 10, residual = 0.00029195248969641466
CG iteration 11, residual = 0.00018574649944941225
CG iteration 12, residual = 0.00013580261400185484
CG iteration 13, residual = 9.190407945869508e-05
CG iteration 14, residual = 5.786867303020079e-05
CG iteration 15, residual = 4.4133664364139676e-05
CG iteration 16, residual = 3.144431877564545e-05
CG iteration 17, residual = 1.9528671630726776e-05
CG iteration 18, residual = 1.3517471299616912e-05
CG iteration 19, residual = 9.890880398831095e-06
CG iteration 20, residual = 7.153565906016266e-06
CG iteration 21, residual = 4.447403561105394e-06
CG iteration 22, residual = 3.1373654614971903e-06
CG iteration 23, residual = 2.128992097134975e-06
CG iteration 24, residual = 1.5382910644371198e-06
CG iteration 25, residual = 1.0783724452556672e-06
CG iteration 26, residual = 7.606999859263063e-07
CG iteration 27, residual = 5.148982458847644e-07
CG iteration 28, residual = 3.360003831578847e-07
CG iteration 29, residual = 2.3316587903829568e-07
CG iteration 30, residual = 1.6242955390947394e-07
CG iteration 31, residual = 1.0751611046927572e-07
CG iteration 32, residual = 7.593395585517017e-08
CG iteration 33, residual = 5.203857345848177e-08
CG iteration 34, residual = 3.596493475563432e-08
CG iteration 35, residual = 2.3081730664357823e-08
CG iteration 36, residual = 1.5146018252583075e-08
CG iteration 37, residual = 1.048575261642813e-08
CG iteration 38, residual = 7.486630162139372e-09
CG iteration 39, residual = 4.871917410683734e-09
CG iteration 40, residual = 3.156959127370897e-09
CG iteration 41, residual = 2.0978635971625215e-09
CG iteration 42, residual = 1.4148898871116803e-09
CG iteration 43, residual = 1.0160441461578859e-09
CG iteration 44, residual = 6.427022488428516e-10
CG iteration 45, residual = 4.4248933637919305e-10
CG iteration 46, residual = 2.916204116607976e-10
CG iteration 47, residual = 1.9313637517747667e-10
CG iteration 48, residual = 1.3914584026564485e-10
CG iteration 49, residual = 9.512926372227085e-11
CG iteration 50, residual = 7.834134799927759e-11
CG iteration 51, residual = 4.9909368671692e-11
CG iteration 52, residual = 3.2056642834272676e-11
CG iteration 53, residual = 2.1830278847414316e-11
CG iteration 54, residual = 1.416601040432362e-11
CG iteration 55, residual = 9.466247168574695e-12
CG iteration 56, residual = 6.343953133208616e-12
CG iteration 57, residual = 4.116199434437733e-12
CG iteration 58, residual = 2.7335529769259413e-12
CG iteration 59, residual = 1.863852973596324e-12
CG iteration 60, residual = 1.273896304199959e-12
CG iteration 61, residual = 8.306910289476918e-13
CG iteration 62, residual = 5.761345876885644e-13
CG iteration 63, residual = 3.9535486980215734e-13
CG iteration 64, residual = 2.484613598156118e-13
CG iteration 65, residual = 1.652428760141027e-13
CG iteration 66, residual = 1.1437371049083882e-13
CG iteration 67, residual = 7.38994192982707e-14
CG iteration 68, residual = 5.3791287529789384e-14
CG iteration 69, residual = 3.874568327134333e-14
CG iteration 70, residual = 2.788097290084638e-14
CG iteration 71, residual = 1.7518143015658498e-14
CG iteration 72, residual = 1.1583744245525836e-14
CG iteration 73, residual = 8.610903250308164e-15
CG iteration 74, residual = 6.286235767250676e-15
CG iteration 75, residual = 4.315913819115871e-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);

[ ]:



[ ]: