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.occ 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
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);

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 = 32897

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, printrates=True)

CG iteration 1, residual = 0.004841340224441174
CG iteration 2, residual = 0.005042736495448021
CG iteration 3, residual = 0.003149747490985734
CG iteration 4, residual = 0.0028071729924236
CG iteration 5, residual = 0.001804428320590604
CG iteration 6, residual = 0.0012102064412063802
CG iteration 7, residual = 0.0009603395596834517
CG iteration 8, residual = 0.0008701989504799061
CG iteration 9, residual = 0.0006325610483462722
CG iteration 10, residual = 0.0004532678408037611
CG iteration 11, residual = 0.00031761673894687684
CG iteration 12, residual = 0.00023160559527484376
CG iteration 13, residual = 0.00020737408603400593
CG iteration 14, residual = 0.0001349457885216425
CG iteration 15, residual = 8.950718248596957e-05
CG iteration 16, residual = 6.491275061094898e-05
CG iteration 17, residual = 8.451296733574526e-05
CG iteration 18, residual = 4.184280949673714e-05
CG iteration 19, residual = 2.896300820833227e-05
CG iteration 20, residual = 1.976331131028852e-05
CG iteration 21, residual = 1.2931661275647855e-05
CG iteration 22, residual = 9.125442581386953e-06
CG iteration 23, residual = 6.593115124205782e-06
CG iteration 24, residual = 4.346076508707978e-06
CG iteration 25, residual = 3.1836366779872845e-06
CG iteration 26, residual = 2.0408071596665487e-06
CG iteration 27, residual = 1.5202327257967885e-06
CG iteration 28, residual = 9.964377451569856e-07
CG iteration 29, residual = 7.321485752707385e-07
CG iteration 30, residual = 1.1250937278504802e-06
CG iteration 31, residual = 4.7674988588792334e-07
CG iteration 32, residual = 3.4051426815080934e-07
CG iteration 33, residual = 3.2838125962618556e-07
CG iteration 34, residual = 1.8700863808042068e-07
CG iteration 35, residual = 1.2816546534517752e-07
CG iteration 36, residual = 8.236163886720542e-08
CG iteration 37, residual = 5.807278174852749e-08
CG iteration 38, residual = 3.836529448570119e-08
CG iteration 39, residual = 2.7371365679727424e-08
CG iteration 40, residual = 1.8847866923971105e-08
CG iteration 41, residual = 1.2744351762808169e-08
CG iteration 42, residual = 9.652152628151719e-09
CG iteration 43, residual = 1.0313815094478265e-08
CG iteration 44, residual = 8.779141698568434e-09
CG iteration 45, residual = 5.487106441851551e-09
CG iteration 46, residual = 3.420018162228465e-09
CG iteration 47, residual = 2.3511731856049253e-09
CG iteration 48, residual = 1.5959262071416761e-09
CG iteration 49, residual = 1.3348905922793637e-09
CG iteration 50, residual = 1.085845101320763e-09
CG iteration 51, residual = 6.190100409201274e-10
CG iteration 52, residual = 4.138812720270018e-10
CG iteration 53, residual = 2.7945479061304556e-10
CG iteration 54, residual = 1.9121230631047716e-10
CG iteration 55, residual = 1.289163445326262e-10
CG iteration 56, residual = 9.053310889681727e-11
CG iteration 57, residual = 1.4616386739201372e-10
CG iteration 58, residual = 6.086166113383403e-11
CG iteration 59, residual = 4.2401063206659764e-11
CG iteration 60, residual = 4.166107594685425e-11
CG iteration 61, residual = 2.2028493658738044e-11
CG iteration 62, residual = 1.4627033263543332e-11
CG iteration 63, residual = 9.555437870747472e-12
CG iteration 64, residual = 6.711632126709203e-12
CG iteration 65, residual = 4.350798219482298e-12
CG iteration 66, residual = 3.181498695871548e-12
CG iteration 67, residual = 2.077153272147553e-12
CG iteration 68, residual = 1.3358956405537314e-12
CG iteration 69, residual = 1.015638462532751e-12
CG iteration 70, residual = 1.4001866800953132e-12
CG iteration 71, residual = 5.407214735677776e-13
CG iteration 72, residual = 3.773441769165803e-13
CG iteration 73, residual = 2.597174685439361e-13
CG iteration 74, residual = 1.664424365724486e-13
CG iteration 75, residual = 1.8374381877922566e-13
CG iteration 76, residual = 9.339367423388565e-14
CG iteration 77, residual = 6.504945649664507e-14
CG iteration 78, residual = 4.3227815015455626e-14
CG iteration 79, residual = 4.4310119945965066e-14
CG iteration 80, residual = 2.5627435026788544e-14
CG iteration 81, residual = 1.622048958509527e-14
CG iteration 82, residual = 1.2266051080827334e-14
CG iteration 83, residual = 1.1529181057455453e-14
CG iteration 84, residual = 1.0883225391715452e-14
CG iteration 85, residual = 6.2379830085508214e-15
CG iteration 86, residual = 5.7261914598780995e-15
CG iteration 87, residual = 3.4197497787012297e-15

Draw (curl(gfu), mesh, "B-field", draw_surf=False, \
      clipping = { "z" : -1, "function":True}, \
      vectors = { "grid_size":50}, min=0, max=2e-5);