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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()

(144582,
 24605,
 ('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: 749548

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 = 23.4089537602072
CG iteration 2, residual = 0.08579555503674476
CG iteration 3, residual = 0.01586869361108836
CG iteration 4, residual = 0.008414263244183094
CG iteration 5, residual = 0.004831031848507708
CG iteration 6, residual = 0.002866998376879903
CG iteration 7, residual = 0.0017001842224325556
CG iteration 8, residual = 0.0010950136771280179
CG iteration 9, residual = 0.0007854402233530557
CG iteration 10, residual = 0.0005749255364233017
CG iteration 11, residual = 0.0004512227180652233
CG iteration 12, residual = 0.0003723743286359982
CG iteration 13, residual = 0.00030286617436472985
CG iteration 14, residual = 0.00023725186653156036
CG iteration 15, residual = 0.00018878268318241247
CG iteration 16, residual = 0.00013412325217794
CG iteration 17, residual = 9.48288618256835e-05
CG iteration 18, residual = 6.590424011379552e-05
CG iteration 19, residual = 4.38259552311419e-05
CG iteration 20, residual = 2.928445829959214e-05
CG iteration 21, residual = 2.0163974115999096e-05
CG iteration 22, residual = 1.3935443189600891e-05
CG iteration 23, residual = 9.647609699861591e-06
CG iteration 24, residual = 6.828740630546225e-06
CG iteration 25, residual = 5.131570470275694e-06
CG iteration 26, residual = 3.794932236793289e-06
CG iteration 27, residual = 2.8666211830974415e-06
CG iteration 28, residual = 2.36614862691119e-06
CG iteration 29, residual = 1.922174806721402e-06
CG iteration 30, residual = 1.5502633305929715e-06
CG iteration 31, residual = 1.208101781312085e-06
CG iteration 32, residual = 8.710042430671508e-07
CG iteration 33, residual = 6.375015106627231e-07
CG iteration 34, residual = 4.4411166763333356e-07
CG iteration 35, residual = 3.200014138785798e-07
CG iteration 36, residual = 2.2195702181758416e-07
CG iteration 37, residual = 1.5505025045802765e-07
CG iteration 38, residual = 1.0535796249740012e-07
CG iteration 39, residual = 7.249863126139948e-08
CG iteration 40, residual = 5.37875246727805e-08
CG iteration 41, residual = 3.7243429932905325e-08
CG iteration 42, residual = 2.579265871793803e-08
CG iteration 43, residual = 1.9548166136252788e-08
CG iteration 44, residual = 1.5510077143907042e-08
CG iteration 45, residual = 1.1831698225777834e-08
CG iteration 46, residual = 9.065014685241804e-09
CG iteration 47, residual = 6.90001430578564e-09
CG iteration 48, residual = 5.312270872998285e-09
CG iteration 49, residual = 3.883357059803542e-09
CG iteration 50, residual = 2.784689182402024e-09
CG iteration 51, residual = 2.0682785760568946e-09
CG iteration 52, residual = 1.565368325633228e-09
CG iteration 53, residual = 1.094012843844026e-09
CG iteration 54, residual = 7.853090896947322e-10
CG iteration 55, residual = 5.547011110856626e-10
CG iteration 56, residual = 3.874516810042643e-10
CG iteration 57, residual = 2.772509361914406e-10
CG iteration 58, residual = 2.0462453124622566e-10
CG iteration 59, residual = 1.5072117753273153e-10
CG iteration 60, residual = 1.0863806234593389e-10
CG iteration 61, residual = 7.96604922094699e-11
CG iteration 62, residual = 6.157075316717481e-11
CG iteration 63, residual = 4.725299458867388e-11
CG iteration 64, residual = 3.302275562305697e-11
CG iteration 65, residual = 2.3429190398887508e-11
CG iteration 66, residual = 1.765602452347201e-11

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