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.

from ngsolve import *
from ngsolve.webgui import Draw
from netgen.csg import *

Geometric model and meshing of a bar magnet:

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

geo = CSGeometry()
geo.Add (air.mat("air"), transparent=True)
geo.Add (magnet.mat("magnet").maxh(1), col=(0.3,0.3,0.1))
# geo.Draw()
mesh = Mesh(geo.GenerateMesh(maxh=2, curvaturesafety=1))
mesh.Curve(3)

<ngsolve.comp.Mesh at 0x7f4a68bea250>

mesh.GetMaterials(), mesh.GetBoundaries()

(('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')

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 = 25068

Assemble system and setup preconditioner using task-parallelization:

with TaskManager():
    a.Assemble()
    f.Assemble()

Finally, declare GridFunction and solve by preconditioned CG iteration:

gfu = GridFunction(fes)
with TaskManager():
    solvers.CG(sol=gfu.vec, rhs=f.vec, mat=a.mat, pre=c.mat)

CG iteration 1, residual = 0.004832539962849041
CG iteration 2, residual = 0.0028801984652913463
CG iteration 3, residual = 0.002263684071776779
CG iteration 4, residual = 0.001859678344364025
CG iteration 5, residual = 0.001370681443141018
CG iteration 6, residual = 0.0009629315328718006
CG iteration 7, residual = 0.0006784248172513822
CG iteration 8, residual = 0.00045373479988693573
CG iteration 9, residual = 0.0003373984523584658
CG iteration 10, residual = 0.00026636366966670494
CG iteration 11, residual = 0.00015785865716709974
CG iteration 12, residual = 0.00010596525810194416
CG iteration 13, residual = 7.440611757876551e-05
CG iteration 14, residual = 5.0217762320984574e-05
CG iteration 15, residual = 3.565457135377302e-05
CG iteration 16, residual = 2.1652775592701034e-05
CG iteration 17, residual = 1.6241088584150837e-05
CG iteration 18, residual = 1.0173867299008845e-05
CG iteration 19, residual = 7.247797857987481e-06
CG iteration 20, residual = 4.782087688901651e-06
CG iteration 21, residual = 3.2977746794455365e-06
CG iteration 22, residual = 2.3147387326143576e-06
CG iteration 23, residual = 1.4401043572085848e-06
CG iteration 24, residual = 1.015420543738993e-06
CG iteration 25, residual = 6.894606759531537e-07
CG iteration 26, residual = 4.933719219535e-07
CG iteration 27, residual = 4.796475553913968e-07
CG iteration 28, residual = 2.9557526024092107e-07
CG iteration 29, residual = 1.839771788491442e-07
CG iteration 30, residual = 1.2576606824444653e-07
CG iteration 31, residual = 8.2123976506946e-08
CG iteration 32, residual = 5.655902360167145e-08
CG iteration 33, residual = 3.71860084719305e-08
CG iteration 34, residual = 2.3956217034615562e-08
CG iteration 35, residual = 1.6498257419541734e-08
CG iteration 36, residual = 1.0759738081310563e-08
CG iteration 37, residual = 7.165063589902991e-09
CG iteration 38, residual = 4.497597039004627e-09
CG iteration 39, residual = 2.915678053894322e-09
CG iteration 40, residual = 1.9460007789523582e-09
CG iteration 41, residual = 1.4015959386010778e-09
CG iteration 42, residual = 8.454869874670649e-10
CG iteration 43, residual = 5.885630548522311e-10
CG iteration 44, residual = 3.7084468751082384e-10
CG iteration 45, residual = 2.5138709989201855e-10
CG iteration 46, residual = 2.0166309107404366e-10
CG iteration 47, residual = 1.7324848959140306e-10
CG iteration 48, residual = 1.2950129899575643e-10
CG iteration 49, residual = 8.738325693900574e-11
CG iteration 50, residual = 5.457581189460101e-11
CG iteration 51, residual = 3.6461818815339004e-11
CG iteration 52, residual = 2.3644971302928238e-11
CG iteration 53, residual = 1.6321141945050875e-11
CG iteration 54, residual = 1.0677540752342985e-11
CG iteration 55, residual = 7.155945040700115e-12
CG iteration 56, residual = 4.796320757795385e-12
CG iteration 57, residual = 3.0989410640522367e-12
CG iteration 58, residual = 2.0500215086040547e-12
CG iteration 59, residual = 1.3892675294535184e-12
CG iteration 60, residual = 8.94075966480835e-13
CG iteration 61, residual = 5.967417873721958e-13
CG iteration 62, residual = 3.903693372245903e-13
CG iteration 63, residual = 2.578190237624934e-13
CG iteration 64, residual = 1.760806427711736e-13
CG iteration 65, residual = 1.1416212949084665e-13
CG iteration 66, residual = 7.617479954176751e-14
CG iteration 67, residual = 6.488758477980287e-14
CG iteration 68, residual = 5.391951885777111e-14
CG iteration 69, residual = 3.032392553262298e-14
CG iteration 70, residual = 2.0506289403518192e-14
CG iteration 71, residual = 1.496355987653864e-14
CG iteration 72, residual = 1.0263477126374197e-14
CG iteration 73, residual = 6.273878808537794e-15
CG iteration 74, residual = 4.1751211823523655e-15

# the vector potential is not supposed to look nice
Draw (gfu, mesh, "vector-potential", draw_surf=False, clipping=True)

Draw (curl(gfu), mesh, "B-field", draw_surf=False, clipping=True)
Draw (1/(mu0*mur)*curl(gfu)-mag, mesh, "H-field", draw_surf=False, clipping=True)

BaseWebGuiScene