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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()

(156257,
 27080,
 ('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: 810392

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 = 23.158149473599167
CG iteration 2, residual = 0.12501570178981108
CG iteration 3, residual = 0.01583997994957571
CG iteration 4, residual = 0.009150865479642162
CG iteration 5, residual = 0.006142161251642189
CG iteration 6, residual = 0.004280697086162398
CG iteration 7, residual = 0.0028989393345321686
CG iteration 8, residual = 0.0021512139863895385
CG iteration 9, residual = 0.0016858412614056178
CG iteration 10, residual = 0.0014516035961544178
CG iteration 11, residual = 0.0012974964190411475
CG iteration 12, residual = 0.0011254345562946237
CG iteration 13, residual = 0.0009329630160477702
CG iteration 14, residual = 0.0007219873758181737
CG iteration 15, residual = 0.0005342404252617543
CG iteration 16, residual = 0.0003914607874518776
CG iteration 17, residual = 0.00027960968063525665
CG iteration 18, residual = 0.00019366829793727368
CG iteration 19, residual = 0.00013067497052168453
CG iteration 20, residual = 9.051937485847922e-05
CG iteration 21, residual = 6.346903741668761e-05
CG iteration 22, residual = 4.69911505304642e-05
CG iteration 23, residual = 3.377602109877608e-05
CG iteration 24, residual = 2.534003885782471e-05
CG iteration 25, residual = 1.9604754098528516e-05
CG iteration 26, residual = 1.5522897316999382e-05
CG iteration 27, residual = 1.3003347018249633e-05
CG iteration 28, residual = 1.0461453977178185e-05
CG iteration 29, residual = 8.896101366135159e-06
CG iteration 30, residual = 7.700229280369582e-06
CG iteration 31, residual = 6.144185537019165e-06
CG iteration 32, residual = 4.58408184974949e-06
CG iteration 33, residual = 3.3849405556326305e-06
CG iteration 34, residual = 2.527087756145998e-06
CG iteration 35, residual = 1.8167508736208374e-06
CG iteration 36, residual = 1.3886166757827448e-06
CG iteration 37, residual = 1.0376014638198029e-06
CG iteration 38, residual = 7.801918593024149e-07
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CG iteration 40, residual = 5.046040783415108e-07
CG iteration 41, residual = 3.8074465091151533e-07
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CG iteration 47, residual = 8.94073008748354e-08
CG iteration 48, residual = 6.816983602516972e-08
CG iteration 49, residual = 4.9489537124945185e-08
CG iteration 50, residual = 3.4998721988218935e-08
CG iteration 51, residual = 2.5790893401006925e-08
CG iteration 52, residual = 1.886631835817722e-08
CG iteration 53, residual = 1.3396797587386309e-08
CG iteration 54, residual = 1.015295624659771e-08
CG iteration 55, residual = 7.640765306563744e-09
CG iteration 56, residual = 5.9486518364397634e-09
CG iteration 57, residual = 4.911964752294718e-09
CG iteration 58, residual = 3.834258756143758e-09
CG iteration 59, residual = 2.992919549532283e-09
CG iteration 60, residual = 2.5037764114110338e-09
CG iteration 61, residual = 2.098479585577693e-09
CG iteration 62, residual = 1.6293104703439676e-09
CG iteration 63, residual = 1.2194226680344694e-09
CG iteration 64, residual = 9.267656128330225e-10
CG iteration 65, residual = 7.826311872400893e-10
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CG iteration 68, residual = 3.005506070922491e-10
CG iteration 69, residual = 2.268269161993438e-10
CG iteration 70, residual = 1.7499670138131875e-10
CG iteration 71, residual = 1.3090263001776984e-10
CG iteration 72, residual = 9.33656635732403e-11
CG iteration 73, residual = 7.043848856208069e-11
CG iteration 74, residual = 5.31818926609895e-11
CG iteration 75, residual = 4.1411395437814484e-11
CG iteration 76, residual = 3.319762600375443e-11
CG iteration 77, residual = 2.6075183860208713e-11
CG iteration 78, residual = 2.1005351429904904e-11

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