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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()

(143558,
 24495,
 ('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: 744342

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 = 23.40691108092222
CG iteration 2, residual = 0.072037148794928
CG iteration 3, residual = 0.012429275073291184
CG iteration 4, residual = 0.008478757871668472
CG iteration 5, residual = 0.004768227510129949
CG iteration 6, residual = 0.002981274148306558
CG iteration 7, residual = 0.0017015420251374211
CG iteration 8, residual = 0.001033597904314153
CG iteration 9, residual = 0.0006530132836549532
CG iteration 10, residual = 0.0004391115547387211
CG iteration 11, residual = 0.0003229553391314064
CG iteration 12, residual = 0.00023156606327798097
CG iteration 13, residual = 0.0001847140896783215
CG iteration 14, residual = 0.00015492108998894758
CG iteration 15, residual = 0.00011803032301800208
CG iteration 16, residual = 8.92758111996088e-05
CG iteration 17, residual = 6.802542369938047e-05
CG iteration 18, residual = 4.636312233867339e-05
CG iteration 19, residual = 3.018512121940736e-05
CG iteration 20, residual = 2.0414433408227176e-05
CG iteration 21, residual = 1.284986174834811e-05
CG iteration 22, residual = 8.375539166432914e-06
CG iteration 23, residual = 6.77329078357961e-06
CG iteration 24, residual = 4.397974980827058e-06
CG iteration 25, residual = 2.640852456712506e-06
CG iteration 26, residual = 1.757358716966491e-06
CG iteration 27, residual = 1.2412038907152507e-06
CG iteration 28, residual = 7.967851794490479e-07
CG iteration 29, residual = 5.600868201872324e-07
CG iteration 30, residual = 3.981401742804579e-07
CG iteration 31, residual = 2.844606348829156e-07
CG iteration 32, residual = 2.1864862196141947e-07
CG iteration 33, residual = 1.6670464014238558e-07
CG iteration 34, residual = 1.2845054732201743e-07
CG iteration 35, residual = 1.0645939933423954e-07
CG iteration 36, residual = 7.79082194555695e-08
CG iteration 37, residual = 6.717721622566847e-08
CG iteration 38, residual = 4.619141164326014e-08
CG iteration 39, residual = 3.013468280957793e-08
CG iteration 40, residual = 2.0213548134386647e-08
CG iteration 41, residual = 1.52157271231662e-08
CG iteration 42, residual = 1.0927612670452739e-08
CG iteration 43, residual = 6.836213746715516e-09
CG iteration 44, residual = 4.645204363305675e-09
CG iteration 45, residual = 3.125241049497664e-09
CG iteration 46, residual = 2.1835079101896105e-09
CG iteration 47, residual = 1.5691074383867186e-09
CG iteration 48, residual = 1.3745922424866552e-09
CG iteration 49, residual = 8.803600598649077e-10
CG iteration 50, residual = 6.042550234998662e-10
CG iteration 51, residual = 4.710018409553864e-10
CG iteration 52, residual = 3.430413682745061e-10
CG iteration 53, residual = 2.5545472929547513e-10
CG iteration 54, residual = 1.873411658833978e-10
CG iteration 55, residual = 1.3349963898016652e-10
CG iteration 56, residual = 9.613951538734398e-11
CG iteration 57, residual = 7.319140071406816e-11
CG iteration 58, residual = 5.45759536972316e-11
CG iteration 59, residual = 4.076871025153753e-11
CG iteration 60, residual = 3.2536671107818956e-11
CG iteration 61, residual = 2.0626181808832435e-11

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