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 = 0.004852219965914332
CG iteration 2, residual = 0.002702030785187379
CG iteration 3, residual = 0.003036764243370699
CG iteration 4, residual = 0.002255860920170469
CG iteration 5, residual = 0.0018096554238315576
CG iteration 6, residual = 0.0014947556366722615
CG iteration 7, residual = 0.0012385853016868154
CG iteration 8, residual = 0.0010288661278606468
CG iteration 9, residual = 0.000951407177638754
CG iteration 10, residual = 0.0006599245626633419
CG iteration 11, residual = 0.0005932100865082764
CG iteration 12, residual = 0.0005039340345763593
CG iteration 13, residual = 0.0003885660382594485
CG iteration 14, residual = 0.00033790160632359435
CG iteration 15, residual = 0.00024105853001483388
CG iteration 16, residual = 0.00018363026806744056
CG iteration 17, residual = 0.0001438841999031142
CG iteration 18, residual = 0.00010694632235333146
CG iteration 19, residual = 8.937901777146266e-05
CG iteration 20, residual = 6.73024114274391e-05
CG iteration 21, residual = 4.585747321701084e-05
CG iteration 22, residual = 3.467426830377006e-05
CG iteration 23, residual = 2.9389740147766077e-05
CG iteration 24, residual = 1.961853749825256e-05
CG iteration 25, residual = 1.6486413455535557e-05
CG iteration 26, residual = 1.1799345138049288e-05
CG iteration 27, residual = 8.662477931805994e-06
CG iteration 28, residual = 7.893962351978612e-06
CG iteration 29, residual = 5.652659766973604e-06
CG iteration 30, residual = 3.858134217102091e-06
CG iteration 31, residual = 2.8096242415922534e-06
CG iteration 32, residual = 2.2428908802732237e-06
CG iteration 33, residual = 1.5926587231740721e-06
CG iteration 34, residual = 1.1947851570006048e-06
CG iteration 35, residual = 8.915180275700868e-07
CG iteration 36, residual = 6.635401090622155e-07
CG iteration 37, residual = 4.899179676394375e-07
CG iteration 38, residual = 3.9389123491363834e-07
CG iteration 39, residual = 4.089638158923769e-07
CG iteration 40, residual = 2.501111704236036e-07
CG iteration 41, residual = 1.8940832068321065e-07
CG iteration 42, residual = 1.5022630134481705e-07
CG iteration 43, residual = 1.5597941827052965e-07
CG iteration 44, residual = 8.980205734457987e-08
CG iteration 45, residual = 7.132447370365771e-08
CG iteration 46, residual = 5.536232534806551e-08
CG iteration 47, residual = 5.379601645822764e-08
CG iteration 48, residual = 3.371561632976866e-08
CG iteration 49, residual = 2.3537101437008038e-08
CG iteration 50, residual = 1.698406314744414e-08
CG iteration 51, residual = 1.227040589953794e-08
CG iteration 52, residual = 9.553586607075795e-09
CG iteration 53, residual = 6.893508549287188e-09
CG iteration 54, residual = 5.6873038709079735e-09
CG iteration 55, residual = 5.194405771178767e-09
CG iteration 56, residual = 3.362719749464923e-09
CG iteration 57, residual = 2.390460654675964e-09
CG iteration 58, residual = 1.760949403770552e-09
CG iteration 59, residual = 1.3050043179874732e-09
CG iteration 60, residual = 9.50325303074194e-10
CG iteration 61, residual = 7.304236834943513e-10
CG iteration 62, residual = 5.396790721153225e-10
CG iteration 63, residual = 5.025920691167629e-10
CG iteration 64, residual = 3.489451208112634e-10
CG iteration 65, residual = 2.5083851228051706e-10
CG iteration 66, residual = 1.7193732461097554e-10
CG iteration 67, residual = 1.2137067179618708e-10
CG iteration 68, residual = 1.261802582127933e-10
CG iteration 69, residual = 8.826648225166917e-11
CG iteration 70, residual = 6.414112791719778e-11
CG iteration 71, residual = 5.7973995300580665e-11
CG iteration 72, residual = 4.1029429185866475e-11
CG iteration 73, residual = 4.148548589620638e-11
CG iteration 74, residual = 2.6130812806690297e-11
CG iteration 75, residual = 1.867042167685853e-11
CG iteration 76, residual = 1.3873735641646724e-11
CG iteration 77, residual = 9.331610386810598e-12
CG iteration 78, residual = 8.770055941614813e-12
CG iteration 79, residual = 7.158897710363732e-12
CG iteration 80, residual = 5.435729690740022e-12
CG iteration 81, residual = 4.396924259407306e-12
CG iteration 82, residual = 2.892654625188373e-12
CG iteration 83, residual = 2.0594670110585923e-12
CG iteration 84, residual = 1.616269723861906e-12
CG iteration 85, residual = 1.3630482166227378e-12
CG iteration 86, residual = 9.32432563153778e-13
CG iteration 87, residual = 6.537727866458235e-13
CG iteration 88, residual = 4.917920249712388e-13
CG iteration 89, residual = 3.6931402476188525e-13
CG iteration 90, residual = 2.4168723659163414e-13
CG iteration 91, residual = 1.965657149778198e-13
CG iteration 92, residual = 1.9448464968292646e-13
CG iteration 93, residual = 1.1991533917981853e-13
CG iteration 94, residual = 1.0959953049428543e-13
CG iteration 95, residual = 7.007006348939615e-14
CG iteration 96, residual = 4.845648770699055e-14
CG iteration 97, residual = 3.496223974304572e-14
CG iteration 98, residual = 3.9195931260975534e-14
CG iteration 99, residual = 2.4740386638142473e-14
CG iteration 100, residual = 1.6108282757783273e-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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