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 = 0.004852219965914344
CG iteration 2, residual = 0.0027020307851873737
CG iteration 3, residual = 0.003036764243370638
CG iteration 4, residual = 0.0022558609201704725
CG iteration 5, residual = 0.001809655423831565
CG iteration 6, residual = 0.001494755636672251
CG iteration 7, residual = 0.0012385853016868033
CG iteration 8, residual = 0.0010288661278606362
CG iteration 9, residual = 0.0009514071776387411
CG iteration 10, residual = 0.0006599245626633309
CG iteration 11, residual = 0.0005932100865082189
CG iteration 12, residual = 0.0005039340345762881
CG iteration 13, residual = 0.0003885660382594456
CG iteration 14, residual = 0.0003379016063236067
CG iteration 15, residual = 0.0002410585300148378
CG iteration 16, residual = 0.00018363026806744257
CG iteration 17, residual = 0.00014388419990311513
CG iteration 18, residual = 0.00010694632235333827
CG iteration 19, residual = 8.93790177714606e-05
CG iteration 20, residual = 6.73024114274383e-05
CG iteration 21, residual = 4.585747321701048e-05
CG iteration 22, residual = 3.467426830377077e-05
CG iteration 23, residual = 2.9389740147766697e-05
CG iteration 24, residual = 1.961853749825276e-05
CG iteration 25, residual = 1.6486413455535726e-05
CG iteration 26, residual = 1.1799345138050608e-05
CG iteration 27, residual = 8.662477931810882e-06
CG iteration 28, residual = 7.893962351984567e-06
CG iteration 29, residual = 5.6526597669714215e-06
CG iteration 30, residual = 3.858134217100691e-06
CG iteration 31, residual = 2.8096242415662105e-06
CG iteration 32, residual = 2.2428908798679096e-06
CG iteration 33, residual = 1.5926587174651224e-06
CG iteration 34, residual = 1.194785035450426e-06
CG iteration 35, residual = 8.915163251244834e-07
CG iteration 36, residual = 6.635059422488201e-07
CG iteration 37, residual = 4.893188148574728e-07
CG iteration 38, residual = 3.8557481040636724e-07
CG iteration 39, residual = 4.0967889660215495e-07
CG iteration 40, residual = 2.518413663037414e-07
CG iteration 41, residual = 1.888017174963876e-07
CG iteration 42, residual = 1.40280768691059e-07
CG iteration 43, residual = 1.1831915672571645e-07
CG iteration 44, residual = 1.0332349527381433e-07
CG iteration 45, residual = 7.295960548902708e-08
CG iteration 46, residual = 6.06313973842733e-08
CG iteration 47, residual = 5.209365202963697e-08
CG iteration 48, residual = 3.314726588808719e-08
CG iteration 49, residual = 2.351973928047344e-08
CG iteration 50, residual = 1.698362724034447e-08
CG iteration 51, residual = 1.2269900752959827e-08
CG iteration 52, residual = 9.546758866682864e-09
CG iteration 53, residual = 6.83045211775255e-09
CG iteration 54, residual = 5.070668086696927e-09
CG iteration 55, residual = 3.984313539055885e-09
CG iteration 56, residual = 3.5239146202048583e-09
CG iteration 57, residual = 2.5662328376945434e-09
CG iteration 58, residual = 1.7687972946726844e-09
CG iteration 59, residual = 1.3065730188335974e-09
CG iteration 60, residual = 9.631038740042897e-10
CG iteration 61, residual = 7.932858920476668e-10
CG iteration 62, residual = 7.02695553251658e-10
CG iteration 63, residual = 4.5847377619936607e-10
CG iteration 64, residual = 3.354277609195442e-10
CG iteration 65, residual = 2.491247828825844e-10
CG iteration 66, residual = 1.7185640874020726e-10
CG iteration 67, residual = 1.2035665262420323e-10
CG iteration 68, residual = 9.721362090491344e-11
CG iteration 69, residual = 9.532241610408825e-11
CG iteration 70, residual = 6.935240393088183e-11
CG iteration 71, residual = 5.7918212177971696e-11
CG iteration 72, residual = 3.7181147557598507e-11
CG iteration 73, residual = 3.075842993896231e-11
CG iteration 74, residual = 3.0133898018571386e-11
CG iteration 75, residual = 1.942021520101536e-11
CG iteration 76, residual = 1.3904875962790232e-11
CG iteration 77, residual = 9.585189088248702e-12
CG iteration 78, residual = 1.0896832304122204e-11
CG iteration 79, residual = 6.6431424430902216e-12
CG iteration 80, residual = 5.132769592820818e-12
CG iteration 81, residual = 4.402212824182317e-12
CG iteration 82, residual = 2.910815520615922e-12
CG iteration 83, residual = 2.0463555223887714e-12
CG iteration 84, residual = 1.5516562840229012e-12
CG iteration 85, residual = 1.229470190638664e-12
CG iteration 86, residual = 9.850121674379977e-13
CG iteration 87, residual = 6.578906274102083e-13
CG iteration 88, residual = 4.923770233159276e-13
CG iteration 89, residual = 3.6994480169038426e-13
CG iteration 90, residual = 2.4541630717915533e-13
CG iteration 91, residual = 2.0728257205542147e-13
CG iteration 92, residual = 1.7028896860273326e-13
CG iteration 93, residual = 1.049184069624091e-13
CG iteration 94, residual = 8.571320835368551e-14
CG iteration 95, residual = 8.384938216649827e-14
CG iteration 96, residual = 4.940890388443257e-14
CG iteration 97, residual = 3.415531306550084e-14
CG iteration 98, residual = 2.6059054653457542e-14
CG iteration 99, residual = 3.025072290185423e-14
CG iteration 100, residual = 1.67910492642488e-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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