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 = 0.004852181316717191
CG iteration 2, residual = 0.0027020320870710757
CG iteration 3, residual = 0.0030377840808697783
CG iteration 4, residual = 0.0022558231750302088
CG iteration 5, residual = 0.0018097230540169032
CG iteration 6, residual = 0.001494200643755157
CG iteration 7, residual = 0.0012388268033693265
CG iteration 8, residual = 0.0010294446699994216
CG iteration 9, residual = 0.0009516097931483282
CG iteration 10, residual = 0.0006604191137835762
CG iteration 11, residual = 0.0005949200300615047
CG iteration 12, residual = 0.0005054392693226587
CG iteration 13, residual = 0.0003888698384542983
CG iteration 14, residual = 0.000338065880998515
CG iteration 15, residual = 0.0002410024706087979
CG iteration 16, residual = 0.00018352671960243655
CG iteration 17, residual = 0.00014395488754058149
CG iteration 18, residual = 0.00010695174779066343
CG iteration 19, residual = 8.938151156459127e-05
CG iteration 20, residual = 6.725501041069648e-05
CG iteration 21, residual = 4.5901465148851605e-05
CG iteration 22, residual = 3.4649313101211516e-05
CG iteration 23, residual = 2.9358296364118494e-05
CG iteration 24, residual = 1.9625013835855123e-05
CG iteration 25, residual = 1.651538087704611e-05
CG iteration 26, residual = 1.1909195395244022e-05
CG iteration 27, residual = 8.969222963267412e-06
CG iteration 28, residual = 8.076287620497462e-06
CG iteration 29, residual = 5.563109692867569e-06
CG iteration 30, residual = 3.843361549385435e-06
CG iteration 31, residual = 2.8067256639299822e-06
CG iteration 32, residual = 2.2394123065548795e-06
CG iteration 33, residual = 1.591497219016032e-06
CG iteration 34, residual = 1.1931279756228445e-06
CG iteration 35, residual = 8.910705555027139e-07
CG iteration 36, residual = 6.636521460339684e-07
CG iteration 37, residual = 4.870393132369188e-07
CG iteration 38, residual = 3.571777007418064e-07
CG iteration 39, residual = 3.0095315120277275e-07
CG iteration 40, residual = 2.850227401506358e-07
CG iteration 41, residual = 1.9797288555081637e-07
CG iteration 42, residual = 1.3948896545114295e-07
CG iteration 43, residual = 9.784842025362251e-08
CG iteration 44, residual = 8.745889280763989e-08
CG iteration 45, residual = 8.355695269913605e-08
CG iteration 46, residual = 5.182668492329859e-08
CG iteration 47, residual = 4.4432073517500434e-08
CG iteration 48, residual = 3.823485722889306e-08
CG iteration 49, residual = 2.378630933288623e-08
CG iteration 50, residual = 1.7024337118407727e-08
CG iteration 51, residual = 1.2345151909046728e-08
CG iteration 52, residual = 9.667779295017163e-09
CG iteration 53, residual = 7.247369158346017e-09
CG iteration 54, residual = 6.884536660553792e-09
CG iteration 55, residual = 4.762856649191651e-09
CG iteration 56, residual = 3.302240627776239e-09
CG iteration 57, residual = 2.3853345147341826e-09
CG iteration 58, residual = 1.765901617904622e-09
CG iteration 59, residual = 1.3076966389351667e-09
CG iteration 60, residual = 9.547317355825023e-10
CG iteration 61, residual = 7.307480414198757e-10
CG iteration 62, residual = 5.251157946308757e-10
CG iteration 63, residual = 4.826393952820061e-10
CG iteration 64, residual = 3.573717369992758e-10
CG iteration 65, residual = 2.5240835667678157e-10
CG iteration 66, residual = 1.717837035542747e-10
CG iteration 67, residual = 1.1978276592667784e-10
CG iteration 68, residual = 8.650913052565187e-11
CG iteration 69, residual = 8.450374161473032e-11
CG iteration 70, residual = 6.628730187286051e-11
CG iteration 71, residual = 5.2719180746986976e-11
CG iteration 72, residual = 4.068554200495785e-11
CG iteration 73, residual = 3.070552792332117e-11
CG iteration 74, residual = 2.6127235502860295e-11
CG iteration 75, residual = 2.1559379421157508e-11
CG iteration 76, residual = 1.3861670883301249e-11
CG iteration 77, residual = 9.192578738551421e-12
CG iteration 78, residual = 7.143506879810661e-12
CG iteration 79, residual = 7.390270392916108e-12
CG iteration 80, residual = 4.773942346442245e-12
CG iteration 81, residual = 3.9414833244326985e-12
CG iteration 82, residual = 3.1725595975610246e-12
CG iteration 83, residual = 2.066345366780213e-12
CG iteration 84, residual = 1.6040176323003826e-12
CG iteration 85, residual = 1.3382952363146786e-12
CG iteration 86, residual = 9.392452559948975e-13
CG iteration 87, residual = 6.429867145561684e-13
CG iteration 88, residual = 4.788941798300373e-13
CG iteration 89, residual = 3.741783138899197e-13
CG iteration 90, residual = 2.545586973304474e-13
CG iteration 91, residual = 2.284842098564151e-13
CG iteration 92, residual = 1.742856271047104e-13
CG iteration 93, residual = 1.4863834523135602e-13
CG iteration 94, residual = 1.0249337267018803e-13
CG iteration 95, residual = 6.797813398715537e-14
CG iteration 96, residual = 4.8306367107962376e-14
CG iteration 97, residual = 3.4109504372517456e-14
CG iteration 98, residual = 2.5251929909954843e-14
CG iteration 99, residual = 2.6422177329978423e-14
CG iteration 100, residual = 1.7807558379408196e-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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