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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()

(107150,
 18459,
 ('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: 556236

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 = 22.997184597229108
CG iteration 2, residual = 0.10464120802136544
CG iteration 3, residual = 0.02037298080951552
CG iteration 4, residual = 0.012357268282558852
CG iteration 5, residual = 0.00604124259380079
CG iteration 6, residual = 0.0034060106455791233
CG iteration 7, residual = 0.0021802523707735127
CG iteration 8, residual = 0.0013424650797709274
CG iteration 9, residual = 0.000976304712097086
CG iteration 10, residual = 0.000750914451894594
CG iteration 11, residual = 0.0005882612416833496
CG iteration 12, residual = 0.0004916508915601537
CG iteration 13, residual = 0.00040496843598494196
CG iteration 14, residual = 0.00028954222937899556
CG iteration 15, residual = 0.00021105079466817825
CG iteration 16, residual = 0.00015016422709833547
CG iteration 17, residual = 9.889048226633663e-05
CG iteration 18, residual = 6.246751522977154e-05
CG iteration 19, residual = 5.019207496404812e-05
CG iteration 20, residual = 3.265468726224532e-05
CG iteration 21, residual = 2.0611442888955747e-05
CG iteration 22, residual = 1.3430819749689912e-05
CG iteration 23, residual = 8.723173325044776e-06
CG iteration 24, residual = 5.640858904737851e-06
CG iteration 25, residual = 3.8128311167996617e-06
CG iteration 26, residual = 2.5439480056769782e-06
CG iteration 27, residual = 1.8945092796356558e-06
CG iteration 28, residual = 1.3972124069369138e-06
CG iteration 29, residual = 1.0334192312920255e-06
CG iteration 30, residual = 7.536270185096147e-07
CG iteration 31, residual = 6.176725716108718e-07
CG iteration 32, residual = 5.252519698853439e-07
CG iteration 33, residual = 4.4534338530791584e-07
CG iteration 34, residual = 3.186964899691976e-07
CG iteration 35, residual = 2.233302446680236e-07
CG iteration 36, residual = 1.5300980351628386e-07
CG iteration 37, residual = 1.0671909410237428e-07
CG iteration 38, residual = 7.458910659221058e-08
CG iteration 39, residual = 4.943890517580285e-08
CG iteration 40, residual = 3.207933316669297e-08
CG iteration 41, residual = 2.078295555158312e-08
CG iteration 42, residual = 1.6206237158580492e-08
CG iteration 43, residual = 1.249554893926422e-08
CG iteration 44, residual = 8.610627649404254e-09
CG iteration 45, residual = 6.058177736319318e-09
CG iteration 46, residual = 3.885832268169193e-09
CG iteration 47, residual = 2.7694221287879868e-09
CG iteration 48, residual = 2.2021739321088924e-09
CG iteration 49, residual = 1.6649704647105492e-09
CG iteration 50, residual = 1.1721800078849162e-09
CG iteration 51, residual = 9.072251631911085e-10
CG iteration 52, residual = 6.874238476485932e-10
CG iteration 53, residual = 5.181209426450161e-10
CG iteration 54, residual = 4.048091207879709e-10
CG iteration 55, residual = 3.2508723418000867e-10
CG iteration 56, residual = 2.3463245538834865e-10
CG iteration 57, residual = 1.6720610965139538e-10
CG iteration 58, residual = 1.137245283764416e-10
CG iteration 59, residual = 7.541177835428416e-11
CG iteration 60, residual = 4.980527832258161e-11
CG iteration 61, residual = 3.173563331208394e-11
CG iteration 62, residual = 2.1936808092126833e-11

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