This page was generated from wta/coil.ipynb.

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 = 23.156764147272217
CG iteration 2, residual = 0.1297365695823336
CG iteration 3, residual = 0.015872725914129207
CG iteration 4, residual = 0.009165557558031081
CG iteration 5, residual = 0.006145351217601874
CG iteration 6, residual = 0.0042742245587745935
CG iteration 7, residual = 0.0028841303153015713
CG iteration 8, residual = 0.002128647485871416
CG iteration 9, residual = 0.00165703074141416
CG iteration 10, residual = 0.001418442578939403
CG iteration 11, residual = 0.0012644251813557473
CG iteration 12, residual = 0.001098510082467778
CG iteration 13, residual = 0.000914410879218985
CG iteration 14, residual = 0.0007114776448350102
CG iteration 15, residual = 0.000529203632401093
CG iteration 16, residual = 0.00038936346712032554
CG iteration 17, residual = 0.0002788773698194183
CG iteration 18, residual = 0.0001934908168673879
CG iteration 19, residual = 0.00013066139088175447
CG iteration 20, residual = 9.050882688088922e-05
CG iteration 21, residual = 6.340998461696633e-05
CG iteration 22, residual = 4.687357131013604e-05
CG iteration 23, residual = 3.360860205515789e-05
CG iteration 24, residual = 2.5131313001257766e-05
CG iteration 25, residual = 1.937135968528969e-05
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CG iteration 27, residual = 1.284615682962333e-05
CG iteration 28, residual = 1.0490643568390958e-05
CG iteration 29, residual = 9.193770173896294e-06
CG iteration 30, residual = 7.648630234352179e-06
CG iteration 31, residual = 5.905124917141659e-06
CG iteration 32, residual = 4.506673137464622e-06
CG iteration 33, residual = 3.4195500277798954e-06
CG iteration 34, residual = 2.697084087081071e-06
CG iteration 35, residual = 2.0852840226680943e-06
CG iteration 36, residual = 1.5524346466972191e-06
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CG iteration 38, residual = 8.446883882090553e-07
CG iteration 39, residual = 6.426225468804017e-07
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CG iteration 47, residual = 9.019193451782849e-08
CG iteration 48, residual = 6.753238912823892e-08
CG iteration 49, residual = 4.9114903478371405e-08
CG iteration 50, residual = 3.4883849719293707e-08
CG iteration 51, residual = 2.5934657606824105e-08
CG iteration 52, residual = 1.9460346244249185e-08
CG iteration 53, residual = 1.495177126830599e-08
CG iteration 54, residual = 1.1581369519446632e-08
CG iteration 55, residual = 8.511792814181706e-09
CG iteration 56, residual = 6.4784465081310525e-09
CG iteration 57, residual = 5.041698621229815e-09
CG iteration 58, residual = 4.088542548044235e-09
CG iteration 59, residual = 3.2820270290936693e-09
CG iteration 60, residual = 2.480411631489129e-09
CG iteration 61, residual = 1.97204111448748e-09
CG iteration 62, residual = 1.5326249370972503e-09
CG iteration 63, residual = 1.1842412610893887e-09
CG iteration 64, residual = 9.084331687860611e-10
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CG iteration 68, residual = 3.2574477114283274e-10
CG iteration 69, residual = 2.493928423532583e-10
CG iteration 70, residual = 1.7564633177104154e-10
CG iteration 71, residual = 1.2819642514351217e-10
CG iteration 72, residual = 9.373243466208206e-11
CG iteration 73, residual = 7.339148830087535e-11
CG iteration 74, residual = 6.043145407008296e-11
CG iteration 75, residual = 4.664979020092712e-11
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CG iteration 77, residual = 2.7518073941984485e-11
CG iteration 78, residual = 2.178888343160176e-11

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