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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()

(154255,
 26582,
 ('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: 799896

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 = 23.402633593987034
CG iteration 2, residual = 0.08803962069870756

CG iteration 3, residual = 0.017000407910615575
CG iteration 4, residual = 0.010511553751425658

CG iteration 5, residual = 0.0065631253609351355
CG iteration 6, residual = 0.004265738148986633

CG iteration 7, residual = 0.0028339339365404917
CG iteration 8, residual = 0.0019799941594800343

CG iteration 9, residual = 0.0015150704408174941
CG iteration 10, residual = 0.0012293671047412683

CG iteration 11, residual = 0.0009973432375400598
CG iteration 12, residual = 0.0008715275423253225

CG iteration 13, residual = 0.0007506181408453045
CG iteration 14, residual = 0.0006003332895921918

CG iteration 15, residual = 0.00047372678901090587
CG iteration 16, residual = 0.00035731611633940425

CG iteration 17, residual = 0.00026620022450004314
CG iteration 18, residual = 0.00020059833840522378

CG iteration 19, residual = 0.00013794270458230654
CG iteration 20, residual = 9.83178613943266e-05

CG iteration 21, residual = 6.956336002656879e-05
CG iteration 22, residual = 4.783117791764149e-05

CG iteration 23, residual = 3.4886354224602264e-05
CG iteration 24, residual = 2.4756017910766987e-05

CG iteration 25, residual = 1.871724009051892e-05
CG iteration 26, residual = 1.4055947314399511e-05

CG iteration 27, residual = 1.069166868931876e-05
CG iteration 28, residual = 8.224223649036217e-06

CG iteration 29, residual = 6.651045227003409e-06
CG iteration 30, residual = 5.673737187235938e-06

CG iteration 31, residual = 4.535212209754745e-06
CG iteration 32, residual = 3.411270557350252e-06

CG iteration 33, residual = 2.780228220057852e-06
CG iteration 34, residual = 2.215379119491946e-06

CG iteration 35, residual = 1.6450237625855566e-06
CG iteration 36, residual = 1.1670791300898314e-06

CG iteration 37, residual = 8.428652978766737e-07
CG iteration 38, residual = 6.39532958865123e-07

CG iteration 39, residual = 4.64274736738767e-07
CG iteration 40, residual = 3.5476379100365915e-07

CG iteration 41, residual = 2.6652732152521576e-07
CG iteration 42, residual = 2.1118018581410372e-07

CG iteration 43, residual = 1.6138284270621408e-07
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CG iteration 45, residual = 1.0437970351118888e-07
CG iteration 46, residual = 8.24694765329773e-08

CG iteration 47, residual = 6.564803433444832e-08
CG iteration 48, residual = 5.463407557403269e-08

CG iteration 49, residual = 4.004239998839463e-08
CG iteration 50, residual = 3.014096633653366e-08

CG iteration 51, residual = 2.256859179027019e-08
CG iteration 52, residual = 1.8082458949834583e-08

CG iteration 53, residual = 1.3579368815067023e-08
CG iteration 54, residual = 1.0079520133621768e-08

CG iteration 55, residual = 7.0888775959719486e-09
CG iteration 56, residual = 5.511326188566599e-09

CG iteration 57, residual = 4.3840544646174615e-09
CG iteration 58, residual = 3.345432879648802e-09

CG iteration 59, residual = 2.6506977387211476e-09
CG iteration 60, residual = 2.128242089463364e-09

CG iteration 61, residual = 1.6574553762779047e-09
CG iteration 62, residual = 1.25391010330175e-09

CG iteration 63, residual = 9.95036868460625e-10
CG iteration 64, residual = 7.510677806947862e-10

CG iteration 65, residual = 5.840196698441732e-10
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CG iteration 67, residual = 3.555582601514917e-10
CG iteration 68, residual = 2.832305251007913e-10

CG iteration 69, residual = 2.2344765140771503e-10
CG iteration 70, residual = 1.636737813478356e-10

CG iteration 71, residual = 1.2664318653236515e-10
CG iteration 72, residual = 9.365102289799813e-11

CG iteration 73, residual = 7.269497192830425e-11
CG iteration 74, residual = 5.896010403380885e-11

CG iteration 75, residual = 4.5035714999345844e-11
CG iteration 76, residual = 3.426725392453175e-11

CG iteration 77, residual = 2.6101415896407014e-11
CG iteration 78, residual = 1.9515872321688652e-11

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