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 = 0.004852219965914341
CG iteration 2, residual = 0.0027020307851873763
CG iteration 3, residual = 0.0030367642433706404
CG iteration 4, residual = 0.0022558609201704734
CG iteration 5, residual = 0.0018096554238315658
CG iteration 6, residual = 0.001494755636672251
CG iteration 7, residual = 0.0012385853016868033
CG iteration 8, residual = 0.0010288661278606364
CG iteration 9, residual = 0.0009514071776387416
CG iteration 10, residual = 0.0006599245626633312
CG iteration 11, residual = 0.0005932100865082195
CG iteration 12, residual = 0.0005039340345762886
CG iteration 13, residual = 0.0003885660382594459
CG iteration 14, residual = 0.00033790160632360655
CG iteration 15, residual = 0.00024105853001483792
CG iteration 16, residual = 0.00018363026806744265
CG iteration 17, residual = 0.00014388419990311519
CG iteration 18, residual = 0.00010694632235333843
CG iteration 19, residual = 8.937901777146065e-05
CG iteration 20, residual = 6.730241142743829e-05
CG iteration 21, residual = 4.585747321701047e-05
CG iteration 22, residual = 3.467426830377074e-05
CG iteration 23, residual = 2.9389740147766697e-05
CG iteration 24, residual = 1.9618537498252772e-05
CG iteration 25, residual = 1.648641345553574e-05
CG iteration 26, residual = 1.1799345138050613e-05
CG iteration 27, residual = 8.662477931810851e-06
CG iteration 28, residual = 7.893962351984486e-06
CG iteration 29, residual = 5.652659766971304e-06
CG iteration 30, residual = 3.858134217099076e-06
CG iteration 31, residual = 2.8096242415200023e-06
CG iteration 32, residual = 2.242890879158173e-06
CG iteration 33, residual = 1.5926587075891235e-06
CG iteration 34, residual = 1.1947848275924221e-06
CG iteration 35, residual = 8.915134511563288e-07
CG iteration 36, residual = 6.634489734392316e-07
CG iteration 37, residual = 4.883287090931206e-07
CG iteration 38, residual = 3.7069322703666377e-07
CG iteration 39, residual = 3.9478398849039317e-07
CG iteration 40, residual = 2.592037826338508e-07
CG iteration 41, residual = 1.8921074923221935e-07
CG iteration 42, residual = 1.3978349465178684e-07
CG iteration 43, residual = 1.1077888010899818e-07
CG iteration 44, residual = 1.0631862911450266e-07
CG iteration 45, residual = 7.347201716837666e-08
CG iteration 46, residual = 5.943952707835033e-08
CG iteration 47, residual = 5.275166242248645e-08
CG iteration 48, residual = 3.32236742540981e-08
CG iteration 49, residual = 2.3521743388482544e-08
CG iteration 50, residual = 1.698369196145712e-08
CG iteration 51, residual = 1.2270255983982614e-08
CG iteration 52, residual = 9.551639146831432e-09
CG iteration 53, residual = 6.875728498865361e-09
CG iteration 54, residual = 5.539500785951503e-09
CG iteration 55, residual = 5.20016889114885e-09
CG iteration 56, residual = 3.3896323580665896e-09
CG iteration 57, residual = 2.3923614326800476e-09
CG iteration 58, residual = 1.7611239856739918e-09
CG iteration 59, residual = 1.3057907770479302e-09
CG iteration 60, residual = 9.59395703735026e-10
CG iteration 61, residual = 7.780780447070185e-10
CG iteration 62, residual = 6.943527932122409e-10
CG iteration 63, residual = 4.633885773609118e-10
CG iteration 64, residual = 3.357246623284069e-10
CG iteration 65, residual = 2.491544721428324e-10
CG iteration 66, residual = 1.718678246603489e-10
CG iteration 67, residual = 1.2048154880130668e-10
CG iteration 68, residual = 9.840288446013254e-11
CG iteration 69, residual = 9.320109961074773e-11
CG iteration 70, residual = 6.868437981461084e-11
CG iteration 71, residual = 5.863273504812311e-11
CG iteration 72, residual = 3.7105836125699176e-11
CG iteration 73, residual = 2.9766104399801836e-11
CG iteration 74, residual = 2.9046067872191827e-11
CG iteration 75, residual = 1.996595140476978e-11
CG iteration 76, residual = 1.3931952760668903e-11
CG iteration 77, residual = 9.755607031928814e-12
CG iteration 78, residual = 1.1247546488426454e-11
CG iteration 79, residual = 6.520237492642227e-12
CG iteration 80, residual = 5.028373409644965e-12
CG iteration 81, residual = 4.42202997842812e-12
CG iteration 82, residual = 2.9224420789981793e-12
CG iteration 83, residual = 2.047258708054821e-12
CG iteration 84, residual = 1.5533271964129212e-12
CG iteration 85, residual = 1.2351980302172217e-12
CG iteration 86, residual = 9.82635539982491e-13
CG iteration 87, residual = 6.577072813900086e-13
CG iteration 88, residual = 4.930529347914644e-13
CG iteration 89, residual = 3.7213593363621184e-13
CG iteration 90, residual = 2.598430617560086e-13
CG iteration 91, residual = 2.2958830657359803e-13
CG iteration 92, residual = 1.6166094353543885e-13
CG iteration 93, residual = 1.0730296182409377e-13
CG iteration 94, residual = 1.158854382732437e-13
CG iteration 95, residual = 7.216308444827935e-14
CG iteration 96, residual = 4.8414739233568314e-14
CG iteration 97, residual = 3.411739645504783e-14
CG iteration 98, residual = 2.55566652558758e-14
CG iteration 99, residual = 2.8569272844122445e-14
CG iteration 100, residual = 1.7241943431997478e-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);

[ ]:



[ ]: