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.csg 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

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)

[2]:

<ngsolve.comp.Mesh at 0x7f5994320f40>

[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 = 25068

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)

CG iteration 1, residual = 0.0048325399628490456
CG iteration 2, residual = 0.0028801984652913476
CG iteration 3, residual = 0.0022636840717767796
CG iteration 4, residual = 0.0018596783443640254
CG iteration 5, residual = 0.0013706814431410186
CG iteration 6, residual = 0.0009629315328718002
CG iteration 7, residual = 0.0006784248172513818
CG iteration 8, residual = 0.00045373479988693513
CG iteration 9, residual = 0.0003373984523584654
CG iteration 10, residual = 0.00026636366966670456
CG iteration 11, residual = 0.00015785865716709944
CG iteration 12, residual = 0.00010596525810194393
CG iteration 13, residual = 7.440611757876529e-05
CG iteration 14, residual = 5.021776232098439e-05
CG iteration 15, residual = 3.565457135377299e-05
CG iteration 16, residual = 2.1652775592700987e-05
CG iteration 17, residual = 1.624108858415077e-05
CG iteration 18, residual = 1.0173867299008301e-05
CG iteration 19, residual = 7.247797857977529e-06
CG iteration 20, residual = 4.782087688678441e-06
CG iteration 21, residual = 3.297774675055772e-06
CG iteration 22, residual = 2.314738643504948e-06
CG iteration 23, residual = 1.440102308370702e-06
CG iteration 24, residual = 1.015364893032002e-06
CG iteration 25, residual = 6.883518518599266e-07
CG iteration 26, residual = 4.7403758426798296e-07
CG iteration 27, residual = 3.88144619137775e-07
CG iteration 28, residual = 3.2959480580885944e-07
CG iteration 29, residual = 1.8514526007224402e-07
CG iteration 30, residual = 1.2579141238184418e-07
CG iteration 31, residual = 8.212444052386796e-08
CG iteration 32, residual = 5.6559033130941875e-08
CG iteration 33, residual = 3.718600863618161e-08
CG iteration 34, residual = 2.3956217036882954e-08
CG iteration 35, residual = 1.649825741956936e-08
CG iteration 36, residual = 1.0759738081270839e-08
CG iteration 37, residual = 7.165063589613023e-09
CG iteration 38, residual = 4.497597037165731e-09
CG iteration 39, residual = 2.9156780336287927e-09
CG iteration 40, residual = 1.9460005896045337e-09
CG iteration 41, residual = 1.4015949518934355e-09
CG iteration 42, residual = 8.454818139627462e-10
CG iteration 43, residual = 5.885193259715897e-10
CG iteration 44, residual = 3.705393301148542e-10
CG iteration 45, residual = 2.483818995237073e-10
CG iteration 46, residual = 1.818774577254239e-10
CG iteration 47, residual = 1.4584906225465505e-10
CG iteration 48, residual = 1.2024723127642808e-10
CG iteration 49, residual = 9.171496865644977e-11
CG iteration 50, residual = 5.570443045402734e-11
CG iteration 51, residual = 3.653030933205259e-11
CG iteration 52, residual = 2.364970790362851e-11
CG iteration 53, residual = 1.6321414195713143e-11
CG iteration 54, residual = 1.0677567762820696e-11
CG iteration 55, residual = 7.155946804501349e-12
CG iteration 56, residual = 4.796320942397389e-12
CG iteration 57, residual = 3.098941072566364e-12
CG iteration 58, residual = 2.050021495652062e-12
CG iteration 59, residual = 1.389267490956996e-12
CG iteration 60, residual = 8.940758432132001e-13
CG iteration 61, residual = 5.967414457254742e-13
CG iteration 62, residual = 3.903682335315413e-13
CG iteration 63, residual = 2.578154878681909e-13
CG iteration 64, residual = 1.7607188882368662e-13
CG iteration 65, residual = 1.1412640666994535e-13
CG iteration 66, residual = 7.569348470518038e-14
CG iteration 67, residual = 5.805577537497316e-14
CG iteration 68, residual = 5.726265827713038e-14
CG iteration 69, residual = 3.030418365664451e-14
CG iteration 70, residual = 1.9899592399072445e-14
CG iteration 71, residual = 1.4080540554146548e-14
CG iteration 72, residual = 1.0298837600879394e-14
CG iteration 73, residual = 6.3275675049075566e-15
CG iteration 74, residual = 4.176805099692409e-15

[7]:

# 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)

[7]:

BaseWebGuiScene

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