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 = 33814

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 = 0.004852219965914336
CG iteration 2, residual = 0.002702030785187379
CG iteration 3, residual = 0.0030367642433706114
CG iteration 4, residual = 0.002255860920170485
CG iteration 5, residual = 0.0018096554238315658
CG iteration 6, residual = 0.0014947556366722591
CG iteration 7, residual = 0.0012385853016868128
CG iteration 8, residual = 0.0010288661278606405
CG iteration 9, residual = 0.0009514071776387368
CG iteration 10, residual = 0.0006599245626633356
CG iteration 11, residual = 0.0005932100865082692
CG iteration 12, residual = 0.0005039340345763445
CG iteration 13, residual = 0.00038856603825944294
CG iteration 14, residual = 0.0003379016063235996
CG iteration 15, residual = 0.00024105853001483602
CG iteration 16, residual = 0.00018363026806744203
CG iteration 17, residual = 0.0001438841999031153
CG iteration 18, residual = 0.00010694632235333537
CG iteration 19, residual = 8.93790177714616e-05
CG iteration 20, residual = 6.730241142743903e-05
CG iteration 21, residual = 4.585747321701098e-05
CG iteration 22, residual = 3.467426830377073e-05
CG iteration 23, residual = 2.9389740147767873e-05
CG iteration 24, residual = 1.9618537498252613e-05
CG iteration 25, residual = 1.6486413455535224e-05
CG iteration 26, residual = 1.179934513804814e-05
CG iteration 27, residual = 8.662477931802176e-06
CG iteration 28, residual = 7.893962351973833e-06
CG iteration 29, residual = 5.65265976697549e-06
CG iteration 30, residual = 3.858134217103924e-06
CG iteration 31, residual = 2.809624241633817e-06
CG iteration 32, residual = 2.2428908809084336e-06
CG iteration 33, residual = 1.592658731992195e-06
CG iteration 34, residual = 1.1947853421887606e-06
CG iteration 35, residual = 8.915205816497372e-07
CG iteration 36, residual = 6.635905934278719e-07
CG iteration 37, residual = 4.907856972475933e-07
CG iteration 38, residual = 4.04782773920426e-07
CG iteration 39, residual = 4.042844169537562e-07
CG iteration 40, residual = 2.48550880744341e-07
CG iteration 41, residual = 1.891581942971264e-07
CG iteration 42, residual = 1.4807578642414037e-07
CG iteration 43, residual = 1.546041972518198e-07
CG iteration 44, residual = 9.050722210403356e-08
CG iteration 45, residual = 7.146460575342831e-08
CG iteration 46, residual = 5.6327689254509866e-08
CG iteration 47, residual = 5.364827733740154e-08
CG iteration 48, residual = 3.354588676261174e-08
CG iteration 49, residual = 2.3531815167319562e-08
CG iteration 50, residual = 1.698392542073839e-08
CG iteration 51, residual = 1.227015866206956e-08
CG iteration 52, residual = 9.550222717278689e-09
CG iteration 53, residual = 6.8626924141819185e-09
CG iteration 54, residual = 5.4194534112999855e-09
CG iteration 55, residual = 5.156887214434929e-09
CG iteration 56, residual = 3.4240739707981977e-09
CG iteration 57, residual = 2.3949957580507545e-09
CG iteration 58, residual = 1.7611183934315407e-09
CG iteration 59, residual = 1.3050877342889607e-09
CG iteration 60, residual = 9.511960628815412e-10
CG iteration 61, residual = 7.357634704904413e-10
CG iteration 62, residual = 5.800147500950869e-10
CG iteration 63, residual = 5.052906205276228e-10
CG iteration 64, residual = 3.4126858381293074e-10
CG iteration 65, residual = 2.497498442729504e-10
CG iteration 66, residual = 1.7191676878806056e-10
CG iteration 67, residual = 1.2136098210795968e-10
CG iteration 68, residual = 1.2006114926964142e-10
CG iteration 69, residual = 9.125152060282384e-11
CG iteration 70, residual = 7.479683955576998e-11
CG iteration 71, residual = 5.3951082373256226e-11
CG iteration 72, residual = 3.723754093441106e-11
CG iteration 73, residual = 3.30364579567858e-11
CG iteration 74, residual = 2.976064237826691e-11
CG iteration 75, residual = 1.9006936655237173e-11
CG iteration 76, residual = 1.389742226053743e-11
CG iteration 77, residual = 9.6739235749913e-12
CG iteration 78, residual = 1.1099581035754909e-11
CG iteration 79, residual = 6.587667022938361e-12
CG iteration 80, residual = 5.189208729616122e-12
CG iteration 81, residual = 4.3826282828174866e-12
CG iteration 82, residual = 2.9088054365811847e-12
CG iteration 83, residual = 2.0711872007114777e-12
CG iteration 84, residual = 1.656669922511515e-12
CG iteration 85, residual = 1.3892668020883356e-12
CG iteration 86, residual = 9.188479720137919e-13
CG iteration 87, residual = 6.532355541295698e-13
CG iteration 88, residual = 4.927701374303353e-13
CG iteration 89, residual = 3.7237124756284523e-13
CG iteration 90, residual = 2.6173437378960135e-13
CG iteration 91, residual = 2.3042540216488633e-13
CG iteration 92, residual = 1.6171686693941402e-13
CG iteration 93, residual = 1.1502139991651479e-13
CG iteration 94, residual = 1.2102683593753146e-13
CG iteration 95, residual = 6.92865936973967e-14
CG iteration 96, residual = 4.838499784939797e-14
CG iteration 97, residual = 3.6167303506657735e-14
CG iteration 98, residual = 4.2871717879109955e-14
CG iteration 99, residual = 2.366176052085044e-14
CG iteration 100, residual = 1.6085373865652154e-14
WARNING: CG did not converge to TOL

[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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