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Magnetostatics

from netgen.occ import *
from ngsolve import *
from ngsolve.webgui import Draw
from netgen.webgui import Draw as DrawGeo
import math

model of the coil:

cyl = Cylinder((0,0,0), Z, r=0.01, h=0.03).faces[0]
heli = Edge(Segment((0,0), (12*math.pi, 0.03)), cyl)
ps = heli.start
vs = heli.start_tangent
pe = heli.end
ve = heli.end_tangent

e1 = Segment((0,0,-0.03), (0,0,-0.01))
c1 = BezierCurve( [(0,0,-0.01), (0,0,0), ps-vs, ps])
e2 = Segment((0,0,0.04), (0,0,0.06))
c2 = BezierCurve( [pe, pe+ve, (0,0,0.03), (0,0,0.04)])
spiral = Wire([e1, c1, heli, c2, e2])
circ = Face(Wire([Circle((0,0,-0.03), Z, 0.001)]))
coil = Pipe(spiral, circ)

coil.faces.maxh=0.2
coil.faces.name="coilbnd"
coil.faces.Max(Z).name="in"
coil.faces.Min(Z).name="out"
coil.mat("coil")
crosssection = coil.faces.Max(Z).mass

DrawGeo (coil);

box = Box((-0.04,-0.04,-0.03), (0.04,0.04,0.06))
box.faces.name = "outer"
air = box-coil
air.mat("air");

mesh-generation of coil and air-box:

geo = OCCGeometry(Glue([coil,air]))
with TaskManager():
    mesh = Mesh(geo.GenerateMesh(meshsize.coarse, maxh=0.01)).Curve(3)

Draw (mesh, clipping={"y":1, "z":0, "dist":0.012});

checking mesh data materials and boundaries:

mesh.ne, mesh.nv, mesh.GetMaterials(), mesh.GetBoundaries()

(154246,
 26567,
 ('coil', 'air'),
 ('out',
  'coilbnd',
  'coilbnd',
  'coilbnd',
  'coilbnd',
  'coilbnd',
  'in',
  'outer',
  'outer',
  'outer',
  'outer',
  'outer',
  'outer'))

Solve a potential problem to determine current density in wire:

on the domain \(\Omega_{\text{coil}}\): \begin{eqnarray*} j & = & \sigma \nabla \Phi \\ \operatorname{div} j & = & 0 \end{eqnarray*} port boundary conditions: \begin{eqnarray*} \Phi & = & 0 \qquad \qquad \text{on } \Gamma_{\text{out}}, \\ j_n & = & \frac{1}{|S|} \quad \qquad \text{on } \Gamma_{\text{in}}, \end{eqnarray*} and \(j_n=0\) else

fespot = H1(mesh, order=3, definedon="coil", dirichlet="out")
phi,psi = fespot.TnT()
sigma = 58.7e6
bfa = BilinearForm(sigma*grad(phi)*grad(psi)*dx).Assemble()
inv = bfa.mat.Inverse(freedofs=fespot.FreeDofs(), inverse="sparsecholesky")
lff = LinearForm(1/crosssection*psi*ds("in")).Assemble()
gfphi = GridFunction(fespot)
gfphi.vec.data = inv * lff.vec

Draw (gfphi, draw_vol=False, clipping={"y":1, "z":0, "dist":0.012});

Solve magnetostatic problem:

current source is current from potential equation:

\[\int \mu^{-1} \operatorname{curl} u \cdot \operatorname{curl} v \, dx = \int j \cdot v \, dx\]

fes = HCurl(mesh, order=2, nograds=True)
print ("HCurl dofs:", fes.ndof)
u,v = fes.TnT()
mu = 4*math.pi*1e-7
a = BilinearForm(1/mu*curl(u)*curl(v)*dx+1e-6/mu*u*v*dx)
pre = Preconditioner(a, "bddc")
f = LinearForm(sigma*grad(gfphi)*v*dx("coil"))
with TaskManager():
    a.Assemble()
    f.Assemble()

HCurl dofs: 799836

from ngsolve.krylovspace import CGSolver
inv = CGSolver(a.mat, pre, printrates=True)
gfu = GridFunction(fes)
with TaskManager():
    gfu.vec.data = inv * f.vec

CG iteration 1, residual = 23.41665955732314
CG iteration 2, residual = 0.07082247543623861

CG iteration 3, residual = 0.01735025821171913
CG iteration 4, residual = 0.011294600452190426

CG iteration 5, residual = 0.007229326867903611
CG iteration 6, residual = 0.004592699361913094

CG iteration 7, residual = 0.003011460554787575
CG iteration 8, residual = 0.0021661342494112934

CG iteration 9, residual = 0.0013910985700330402
CG iteration 10, residual = 0.0010247886911365894

CG iteration 11, residual = 0.0007805692940897332
CG iteration 12, residual = 0.000585695304715673

CG iteration 13, residual = 0.00047216897897238143
CG iteration 14, residual = 0.00039075228129527355

CG iteration 15, residual = 0.0003317695459404256
CG iteration 16, residual = 0.0002807740328751362

CG iteration 17, residual = 0.0002283392866292569
CG iteration 18, residual = 0.00017630481162271845

CG iteration 19, residual = 0.00013261173737393584
CG iteration 20, residual = 0.00010191963360324295

CG iteration 21, residual = 7.349563352014575e-05
CG iteration 22, residual = 5.6585916822506634e-05

CG iteration 23, residual = 4.031527088633962e-05
CG iteration 24, residual = 2.9310557716221545e-05

CG iteration 25, residual = 2.1517654396119172e-05
CG iteration 26, residual = 1.5315007819277447e-05

CG iteration 27, residual = 1.1265791512891458e-05
CG iteration 28, residual = 8.764470970074293e-06

CG iteration 29, residual = 7.006903720853185e-06
CG iteration 30, residual = 6.11810641382005e-06

CG iteration 31, residual = 5.187755439966227e-06
CG iteration 32, residual = 4.296791580657649e-06

CG iteration 33, residual = 3.395777002036293e-06
CG iteration 34, residual = 2.6537132149389263e-06

CG iteration 35, residual = 2.085001068277323e-06
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CG iteration 37, residual = 1.2032959499067467e-06
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CG iteration 39, residual = 6.593978538512096e-07
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CG iteration 41, residual = 3.7340692852849274e-07
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CG iteration 43, residual = 2.519568142559646e-07
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CG iteration 45, residual = 1.6751968855857867e-07
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CG iteration 47, residual = 1.0794192723162264e-07
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CG iteration 49, residual = 6.95031435671563e-08
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CG iteration 51, residual = 4.186623778535872e-08
CG iteration 52, residual = 3.0758435677572776e-08

CG iteration 53, residual = 2.282387259697462e-08
CG iteration 54, residual = 1.659595703123311e-08

CG iteration 55, residual = 1.28649311755385e-08
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CG iteration 57, residual = 8.35009994749917e-09
CG iteration 58, residual = 6.031988060506761e-09

CG iteration 59, residual = 5.0399542443946075e-09
CG iteration 60, residual = 4.281296161672665e-09

CG iteration 61, residual = 3.183696969083653e-09
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CG iteration 63, residual = 2.113504744340369e-09
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CG iteration 65, residual = 1.2651438801607538e-09
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CG iteration 67, residual = 7.654592723763657e-10
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CG iteration 69, residual = 4.490020695778679e-10
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CG iteration 71, residual = 2.63353785123771e-10
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CG iteration 73, residual = 1.6068248466615064e-10
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CG iteration 75, residual = 9.998933397158826e-11
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CG iteration 77, residual = 5.562410556432552e-11
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CG iteration 79, residual = 3.584955285885154e-11
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CG iteration 81, residual = 2.2335791453544646e-11

Draw (curl(gfu), mesh, draw_surf=False, \
      min=0, max=3e-4, clipping = { "y":1, "z" : 0, "function":False}, vectors = { "grid_size":100});