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 \(M\):
Introducing a vector-potential \(A\) such that \(B = \Curl A\), and putting equations together we get
In weak form: Find \(A \in H(\Curl)\) such that
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 = 32897
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 = 0.004841340224441175
CG iteration 2, residual = 0.00504273649544808
CG iteration 3, residual = 0.0031497474909857243
CG iteration 4, residual = 0.0028071729924236356
CG iteration 5, residual = 0.001804428320590585
CG iteration 6, residual = 0.001210206441206369
CG iteration 7, residual = 0.0009603395596834489
CG iteration 8, residual = 0.000870198950479886
CG iteration 9, residual = 0.0006325610483462757
CG iteration 10, residual = 0.0004532678408037579
CG iteration 11, residual = 0.00031761673894694563
CG iteration 12, residual = 0.00023160559528270013
CG iteration 13, residual = 0.0002073740862753844
CG iteration 14, residual = 0.00013494579642198826
CG iteration 15, residual = 8.950783625998786e-05
CG iteration 16, residual = 6.498833461086092e-05
CG iteration 17, residual = 8.616619984193018e-05
CG iteration 18, residual = 4.163212489626196e-05
CG iteration 19, residual = 2.8962061845258242e-05
CG iteration 20, residual = 1.9763306911030648e-05
CG iteration 21, residual = 1.293166126174068e-05
CG iteration 22, residual = 9.125442581344633e-06
CG iteration 23, residual = 6.593115124197943e-06
CG iteration 24, residual = 4.346076508598786e-06
CG iteration 25, residual = 3.1836366768641896e-06
CG iteration 26, residual = 2.040807143144261e-06
CG iteration 27, residual = 1.5202328505901764e-06
CG iteration 28, residual = 9.964826861291812e-07
CG iteration 29, residual = 7.376541614315532e-07
CG iteration 30, residual = 1.148967262523561e-06
CG iteration 31, residual = 4.7319994787207325e-07
CG iteration 32, residual = 3.359824295824361e-07
CG iteration 33, residual = 3.304117997390097e-07
CG iteration 34, residual = 1.8737442944835216e-07
CG iteration 35, residual = 1.281800722916626e-07
CG iteration 36, residual = 8.236192812293856e-08
CG iteration 37, residual = 5.807279735402932e-08
CG iteration 38, residual = 3.836540671410121e-08
CG iteration 39, residual = 2.7372896303574583e-08
CG iteration 40, residual = 1.886439131458214e-08
CG iteration 41, residual = 1.2940810041258539e-08
CG iteration 42, residual = 1.0949566826135906e-08
CG iteration 43, residual = 9.70292171259168e-09
CG iteration 44, residual = 1.0538452624843142e-08
CG iteration 45, residual = 5.136909745239321e-09
CG iteration 46, residual = 3.4083419424946384e-09
CG iteration 47, residual = 2.4538296049859e-09
CG iteration 48, residual = 2.2742843524323708e-09
CG iteration 49, residual = 1.4219421981513053e-09
CG iteration 50, residual = 9.616740028193906e-10
CG iteration 51, residual = 6.138968159498361e-10
CG iteration 52, residual = 4.137534619729871e-10
CG iteration 53, residual = 2.794512559159817e-10
CG iteration 54, residual = 1.9121247382912852e-10
CG iteration 55, residual = 1.2894989126831692e-10
CG iteration 56, residual = 9.572856552537582e-11
CG iteration 57, residual = 1.383630427650971e-10
CG iteration 58, residual = 6.018814213914598e-11
CG iteration 59, residual = 4.415747790421181e-11
CG iteration 60, residual = 4.059329957016689e-11
CG iteration 61, residual = 2.1958818735334824e-11
CG iteration 62, residual = 1.4625600065585043e-11
CG iteration 63, residual = 9.555414431457189e-12
CG iteration 64, residual = 6.711631580705357e-12
CG iteration 65, residual = 4.350798210560002e-12
CG iteration 66, residual = 3.1814987397268173e-12
CG iteration 67, residual = 2.0771587021256846e-12
CG iteration 68, residual = 1.3370176759318846e-12
CG iteration 69, residual = 1.1787674040817758e-12
CG iteration 70, residual = 1.1580099718132047e-12
CG iteration 71, residual = 5.395689595531632e-13
CG iteration 72, residual = 3.779361176007076e-13
CG iteration 73, residual = 2.6751962040513e-13
CG iteration 74, residual = 2.432392309349956e-13
CG iteration 75, residual = 1.5294699892543688e-13
CG iteration 76, residual = 9.106238698874907e-14
CG iteration 77, residual = 6.767945737456721e-14
CG iteration 78, residual = 6.404792237244913e-14
CG iteration 79, residual = 3.789878217167412e-14
CG iteration 80, residual = 2.475906671579798e-14
CG iteration 81, residual = 1.624474299448155e-14
CG iteration 82, residual = 1.2447905633164482e-14
CG iteration 83, residual = 1.2268736711654367e-14
CG iteration 84, residual = 1.0475448413338389e-14
CG iteration 85, residual = 6.5503771202162144e-15
CG iteration 86, residual = 5.744001798118553e-15
CG iteration 87, residual = 3.3689491771466907e-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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