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 = 33832

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 = 0.004821720077892097
CG iteration 2, residual = 0.0032104516402596925
CG iteration 3, residual = 0.0023715429106491766
CG iteration 4, residual = 0.0016322043090891188
CG iteration 5, residual = 0.0012442854977395641
CG iteration 6, residual = 0.0009044032351893659
CG iteration 7, residual = 0.0007001208867136665
CG iteration 8, residual = 0.0005608532265804454
CG iteration 9, residual = 0.0003638790580497855
CG iteration 10, residual = 0.0002919524896964143
CG iteration 11, residual = 0.0001857464994494118
CG iteration 12, residual = 0.00013580261400185457
CG iteration 13, residual = 9.190407945869473e-05
CG iteration 14, residual = 5.786867303020054e-05
CG iteration 15, residual = 4.413366436413943e-05
CG iteration 16, residual = 3.144431877564533e-05
CG iteration 17, residual = 1.95286716307267e-05
CG iteration 18, residual = 1.351747129961686e-05
CG iteration 19, residual = 9.890880398831081e-06
CG iteration 20, residual = 7.153565906016231e-06
CG iteration 21, residual = 4.447403561105373e-06
CG iteration 22, residual = 3.137365461497177e-06
CG iteration 23, residual = 2.1289920971349657e-06
CG iteration 24, residual = 1.5382910644371123e-06
CG iteration 25, residual = 1.0783724452556627e-06
CG iteration 26, residual = 7.606999859263041e-07
CG iteration 27, residual = 5.148982458847627e-07
CG iteration 28, residual = 3.3600038315788313e-07
CG iteration 29, residual = 2.3316587903829451e-07
CG iteration 30, residual = 1.624295539094731e-07
CG iteration 31, residual = 1.0751611046927506e-07
CG iteration 32, residual = 7.593395585516968e-08
CG iteration 33, residual = 5.203857345848136e-08
CG iteration 34, residual = 3.596493475563404e-08
CG iteration 35, residual = 2.308173066435777e-08
CG iteration 36, residual = 1.5146018252584276e-08
CG iteration 37, residual = 1.0485752616436176e-08
CG iteration 38, residual = 7.486630162168182e-09
CG iteration 39, residual = 4.871917410790807e-09
CG iteration 40, residual = 3.1569591281058936e-09
CG iteration 41, residual = 2.0978636033785596e-09
CG iteration 42, residual = 1.4148899417401516e-09
CG iteration 43, residual = 1.0160443743397859e-09
CG iteration 44, residual = 6.427034242217112e-10
CG iteration 45, residual = 4.4249734055270333e-10
CG iteration 46, residual = 2.916680574546915e-10
CG iteration 47, residual = 1.9351423031660779e-10
CG iteration 48, residual = 1.4096584987468845e-10
CG iteration 49, residual = 1.0038039080232728e-10
CG iteration 50, residual = 8.06999854206486e-11
CG iteration 51, residual = 4.8904484505713945e-11
CG iteration 52, residual = 3.19891806052492e-11
CG iteration 53, residual = 2.1825110020424868e-11
CG iteration 54, residual = 1.4165656492627324e-11
CG iteration 55, residual = 9.466227235675367e-12
CG iteration 56, residual = 6.34395195287346e-12
CG iteration 57, residual = 4.116199372813897e-12
CG iteration 58, residual = 2.7335529769319452e-12
CG iteration 59, residual = 1.863852989325787e-12
CG iteration 60, residual = 1.2738963637682076e-12
CG iteration 61, residual = 8.306912941053121e-13
CG iteration 62, residual = 5.761357442337578e-13
CG iteration 63, residual = 3.953586871163145e-13
CG iteration 64, residual = 2.4847768353414647e-13
CG iteration 65, residual = 1.6537342155921084e-13
CG iteration 66, residual = 1.1486317861857926e-13
CG iteration 67, residual = 7.631251054593411e-14
CG iteration 68, residual = 5.920525612276733e-14
CG iteration 69, residual = 4.039637068995113e-14
CG iteration 70, residual = 2.754534107737246e-14
CG iteration 71, residual = 1.7534686091384046e-14
CG iteration 72, residual = 1.1817661315929153e-14
CG iteration 73, residual = 8.983968802374673e-15
CG iteration 74, residual = 6.335203111976815e-15
CG iteration 75, residual = 4.266932949531714e-15

[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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