C型電磁石がつくる磁場¶

C型電磁石が作る磁場の解析結果を以下に示す．

問題設定 / メッシュ¶

C型の鉄芯を考える．鉄芯の断面は長さ 0.2 (m) の正方形とし、鉄芯の中心経路は一辺 0.5 (m) とする（つまり、C型電磁石の一辺の合計長さは 0.1+0.5+0.1 = 0.7 (m) となる）．
中心経路をつくる正方形のうち、ギャップと反対側の辺にコイルを設置する．コイルはコーナー部を除いた鉄芯直線部分のうち、60 (%) の領域に巻きつける．直線部分は長さ 0.5-0.1-0.1 = 0.3 であるからコイルの巻きつけ長さは 0.3*0.6 = 0.18 であり、コイル幅は 0.02 (m) と設定する．コイル電流は 100 kA とする．
計算領域は、1辺 2.0 (m) の立方体の計算空間を考え、境界で電磁ポテンシャルは Dirichlet 条件 $A_\phi=0$ を課す．

../../../_images/cshape_magnet_model.png

C型電磁石のメッシュ生成プログラム¶

メッシュ生成用プログラムを以下に示す．

C型電磁石モデルのメッシュ生成用 gmsh-API python プログラム¶

import numpy as np
import os, sys
import gmsh_api.gmsh as gmsh

import nkGmshRoutines.generate__quadShape  as qua
import nkGmshRoutines.generate__squareTube as tub


# ------------------------------------------------- #
# --- [1] initialization of the gmsh            --- #
# ------------------------------------------------- #
gmsh.initialize()
gmsh.option.setNumber( "General.Terminal", 1 )
gmsh.model.add( "model" )

# ------------------------------------------------- #
# --- [2] initialize settings                   --- #
# ------------------------------------------------- #
ptsDim , lineDim , surfDim , voluDim  =  0,  1,  2,  3
pts    , line    , surf    , volu     = {}, {}, {}, {}
ptsPhys, linePhys, surfPhys, voluPhys = {}, {}, {}, {}
x_, y_, z_, lc_, tag_                 =  0,  1,  2,  3, 4

# ------------------------------------------------- #
# --- [3] Modeling                              --- #
# ------------------------------------------------- #

lc_sim     =  0.800
lc_mag     =  0.010
lc_coil    =  0.010

gap        =  0.100
Lmag       =  0.500
Lsquare    =  0.200
coil_width =  0.020

# ------------------------------------------------- #
# --- C-shaped Magnet core                      --- #
# ------------------------------------------------- #

Ledge_out  =   0.5*Lmag + 0.5*Lsquare
Ledge_inn  =   0.5*Lmag - 0.5*Lsquare
Lhalf_gap  =   0.5*gap
y_minus    = - 0.5*Lsquare

pts["x1" ] = [ -Ledge_out, y_minus, -Ledge_out, lc_mag, 0 ]
pts["x2" ] = [ +Ledge_out, y_minus, -Ledge_out, lc_mag, 0 ]
pts["x3" ] = [ +Ledge_out, y_minus, -Lhalf_gap, lc_mag, 0 ]
pts["x4" ] = [ +Ledge_inn, y_minus, -Lhalf_gap, lc_mag, 0 ]
pts["x5" ] = [ +Ledge_inn, y_minus, -Ledge_inn, lc_mag, 0 ]
pts["x6" ] = [ -Ledge_inn, y_minus, -Ledge_inn, lc_mag, 0 ]
pts["x7" ] = [ -Ledge_inn, y_minus, +Ledge_inn, lc_mag, 0 ]
pts["x8" ] = [ +Ledge_inn, y_minus, +Ledge_inn, lc_mag, 0 ]
pts["x9" ] = [ +Ledge_inn, y_minus, +Lhalf_gap, lc_mag, 0 ]
pts["x10"] = [ +Ledge_out, y_minus, +Lhalf_gap, lc_mag, 0 ]
pts["x11"] = [ +Ledge_out, y_minus, +Ledge_out, lc_mag, 0 ]
pts["x12"] = [ -Ledge_out, y_minus, +Ledge_out, lc_mag, 0 ]

# -- add points -- #
for key in list( pts.keys() ):
    pts[key][4] = gmsh.model.occ.addPoint( pts[key][0], pts[key][1], pts[key][2], meshSize=pts[key][3] )

# -- add lines  -- #
lineLoop = []
index    = [ ik+1 for ik in range(12) ] + [1]
for ik in range( len( index )-1 ):
    ik1, ik2       = index[ik], index[ik+1]
    ptkey1, ptkey2 = "x{0}".format(ik1), "x{0}".format(ik2)
    linekey        = "line_{0}_{1}".format(ik1,ik2)
    line[linekey]  = gmsh.model.occ.addLine( pts[ptkey1][tag_], pts[ptkey2][tag_] )
    lineLoop.append( line[linekey] )

# -- add surface  -- #
lineGroup          = gmsh.model.occ.addCurveLoop( lineLoop )
surf["cMag"]       = gmsh.model.occ.addPlaneSurface( [ lineGroup ] )

# -- add volume   -- #
delta              = [ 0.0, Lsquare, 0.0 ]
ret1               = gmsh.model.occ.extrude( [ (surfDim,surf["cMag"]) ], delta[0], delta[1], delta[2] )
volu["cMag"]       = ret1[1][1]


# ------------------------------------------------- #
# --- coil around magnetic core                 --- #
# ------------------------------------------------- #

coil_length  = ( Lmag - Lsquare ) * 0.6
xcent        = - 0.5*Lmag
ycent        =   0.0
d_in         =   0.5*Lsquare
d_out        =   0.5*Lsquare + coil_width
z_bot        = - 0.5*coil_length
z_top        = + 0.5*coil_length

vL1_coil_1   = [ xcent-d_out, ycent-d_out, z_bot ]
vL1_coil_2   = [ xcent-d_in , ycent-d_out, z_bot ]
vL1_coil_3   = [ xcent-d_in , ycent-d_out, z_top ]
vL1_coil_4   = [ xcent-d_out, ycent-d_out, z_top ]
delta1       = [         0.0,   2.0*d_out,   0.0 ]

vL2_coil_1   = [ xcent-d_in , ycent-d_out, z_bot ]
vL2_coil_2   = [ xcent+d_in , ycent-d_out, z_bot ]
vL2_coil_3   = [ xcent+d_in , ycent-d_out, z_top ]
vL2_coil_4   = [ xcent-d_in , ycent-d_out, z_top ]
delta2       = [         0.0,  coil_width,   0.0 ]

vL3_coil_1   = [ xcent-d_in , ycent+d_in , z_bot ]
vL3_coil_2   = [ xcent+d_in , ycent+d_in , z_bot ]
vL3_coil_3   = [ xcent+d_in , ycent+d_in , z_top ]
vL3_coil_4   = [ xcent-d_in , ycent+d_in , z_top ]
delta3       = [         0.0,  coil_width,   0.0 ]

vL4_coil_1   = [ xcent+d_in , ycent-d_out, z_bot ]
vL4_coil_2   = [ xcent+d_out, ycent-d_out, z_bot ]
vL4_coil_3   = [ xcent+d_out, ycent-d_out, z_top ]
vL4_coil_4   = [ xcent+d_in , ycent-d_out, z_top ]
delta4       = [         0.0,   2.0*d_out,   0.0 ]

ret2_1       = qua.generate__quadShape( lc=lc_coil, extrude_delta=delta1, defineVolu=True, \
                                        x1=vL1_coil_1, x2=vL1_coil_2, \
                                        x3=vL1_coil_3, x4=vL1_coil_4  )
ret2_2       = qua.generate__quadShape( lc=lc_coil, extrude_delta=delta2, defineVolu=True, \
                                        x1=vL2_coil_1, x2=vL2_coil_2, \
                                        x3=vL2_coil_3, x4=vL2_coil_4  )
ret2_3       = qua.generate__quadShape( lc=lc_coil, extrude_delta=delta3, defineVolu=True, \
                                        x1=vL3_coil_1, x2=vL3_coil_2, \
                                        x3=vL3_coil_3, x4=vL3_coil_4  )
ret2_4       = qua.generate__quadShape( lc=lc_coil, extrude_delta=delta4, defineVolu=True, \
                                        x1=vL4_coil_1, x2=vL4_coil_2, \
                                        x3=vL4_coil_3, x4=vL4_coil_4  )
volu["coil1"] = ret2_1["volu"]["quad"]
volu["coil2"] = ret2_2["volu"]["quad"]
volu["coil3"] = ret2_3["volu"]["quad"]
volu["coil4"] = ret2_4["volu"]["quad"]

# ------------------------------------------------- #
# --- simulation region                         --- #
# ------------------------------------------------- #

simulation_region = "box"

if ( simulation_region == "box" ):
    xSim    = 2.0
    ySim    = 2.0
    zSim    = 2.0

    x1_sim  = [ -xSim, -ySim,    -zSim ]
    x2_sim  = [ -xSim, +ySim,    -zSim ]
    x3_sim  = [ +xSim, +ySim,    -zSim ]
    x4_sim  = [ +xSim, -ySim,    -zSim ]
    delta   = [   0.0,   0.0, 2.0*zSim ]
    ret3    = qua.generate__quadShape( lc=lc_sim, x1=x1_sim, x2=x2_sim, x3=x3_sim, x4=x4_sim, \
                                       extrude_delta=delta, defineVolu=True )
    volu["air"] = ret3["volu"]["quad"]

elif ( simulation_region == "sphere" ):
    sph     = [ 0.0, 0.0, 0.0, 2.0 ]
    ret     = gmsh.model.occ.addSphere( sph[0], sph[1], sph[2], sph[3] )
    volu["air"] = ret

    
# ------------------------------------------------- #
# --- [4] Physical Grouping                     --- #
# ------------------------------------------------- #

gmsh.model.occ.removeAllDuplicates()

farBCs  = [ 28, 29, 30, 31, 32, 33 ]

gmsh.model.occ.synchronize()
voluPhys["cMag"] = gmsh.model.addPhysicalGroup( voluDim, [ volu["cMag"] ], tag=301 )
voluPhys["coil"] = gmsh.model.addPhysicalGroup( voluDim, [ volu["coil1"], volu["coil2"], \
                                                           volu["coil3"], volu["coil4"] ], tag=302 )
voluPhys["air"]  = gmsh.model.addPhysicalGroup( voluDim, [ volu["air"]  ], tag=303 )

surfPhys["far"]  = gmsh.model.addPhysicalGroup( surfDim, farBCs, tag=201 )

# ------------------------------------------------- #
# --- [2] post process                          --- #
# ------------------------------------------------- #
gmsh.model.occ.synchronize()
gmsh.model.mesh.generate(3)
gmsh.write( "model.geo_unrolled" )
gmsh.write( "model.msh" )
gmsh.finalize()

C型電磁石の磁場解析用 Elmer入力ファイル¶

以下にElmer入力ファイルのサンプルを示す．

C型電磁石がつくる磁場の Elmer 入力ファイル ( Cshape_magnet.sif )¶

! ========================================================= !
! ===  C-Shape Magnet and coil                          === !
! ========================================================= !

! ------------------------------------------------- !
! --- [1] Global Simulation Settings            --- !
! ------------------------------------------------- !

CHECK KEYWORDS "Warn"

Header
  Mesh DB "." "model"
  Include Path ""
  Results Directory ""
End

Simulation
  coordinate system           = "Cartesian"
  Coordinate Mapping(3)       = 1 2 3

  Simulation Type             = "Steady State"
  Steady State Max Iterations = 1
  
  Solver Input File           = "cShape_magnet.sif"
  Output File                 = "cShape_magnet.dat"
  Post File                   = "cShape_magnet.vtu"
End


Constants
  Permeability of Vacuum = 1.2566e-06
End


! ------------------------------------------------- !
! --- [2] Body & Material Settings              --- !
! ------------------------------------------------- !

Body 1
  Target Bodies(1) = 302
  Name             = "coil"

  Equation         = 1
  Material         = 1
  Body Force       = 1
End

Body 2
  Target Bodies(1) = 301
  Name             = "core"

  Equation         = 2
  Material         = 2
End

Body 3
  Target Bodies(1) = 303
  Name             = "air"

  Equation         = 2
  Material         = 3
End


Material 1
  Name                  = "Cupper"
  Electric Conductivity = 5.0e7
  Relative Permittivity = 1.0
  Relative Permeability = 1.0
End

Material 2
  Name                  = "Iron"
  Electric Conductivity = 1.0e7
  Relative Permittivity = 1.0
  Relative Permeability = 5000

  H-B Curve(38,2) = Real
      INCLUDE HBcurve.dat
End

Material 3
  Name                  = "Air"
  Electric Conductivity = 0.0
  Relative Permittivity = 1.0
  Relative Permeability = 1.0
End


Component 1
  Name                    = "Coil"
  Coil Type               = "test"
  Master Bodies(1)        = 1
  Desired Coil Current    = +1.0e5
End

! ------------------------------------------------- !
! --- [3] Equation & Solver Settings            --- !
! ------------------------------------------------- !

Equation 1
  Name              = "MagneticField_in_Coil"
  Active Solvers(3) = 1 2 3
End

Equation 2
  Name              = "MagneticField_in_Material"
  Active Solvers(2) = 2 3
End


Solver 1
  Equation                            = "CoilSolver"
  Procedure                           = "CoilSolver" "CoilSolver"

  Linear System Solver                = "Iterative"
  Linear System Preconditioning       = "ILU1"
  Linear System Max Iterations        = 3000
  Linear System Convergence Tolerance = 1e-08
  Linear System Iterative Method      = "BiCGStabL"
  Linear System Residual Output       = 20
  Steady State Convergence Tolerance  = 1e-06

  Optimize Bandwidth                  = True
  Nonlinear System Consistent Norm    = True
  Coil Closed                         = Logical True
  Narrow Interface                    = Logical False

  Normalize Coil Current              = Logical True
  Save Coil Set                       = Logical False
  Save Coil Index                     = Logical False
  Calculate Elemental Fields          = Logical True
End

Solver 2
  Equation                            = "WhitneySolver"
  Variable                            = "AV"
  Variable Dofs                       = 1
  Procedure                           = "MagnetoDynamics" "WhitneyAVSolver"

  Linear System Solver                = "Iterative"
  Linear System Iterative Method      = "BiCGStab"
  Linear System Max Iterations        = 3000
  Linear System Convergence Tolerance = 1.0e-6
  Linear System Preconditioning       = "None"
  Linear System Symmetric             = True

  Steady State Convergence Tolerance       = 1.0e-6
  Nonlinear System Convergence Tolerance   = 1.0e-8
  Nonlinear System Max Iterations          = 100
  Nonlinear System Newton After Iterations = 3
  Nonlinear System Newton After Tolerance  = 1.0e-8
  Nonlinear System Relaxation Factor       = 0.5
End

Solver 3
  Equation                            = "MGDynamicsCalc"
  Procedure                           = "MagnetoDynamics" "MagnetoDynamicsCalcFields"
  Potential Variable                  = String "AV"

  Calculate Current Density           = Logical True
  Calculate Magnetic Field Strength   = Logical True

  Steady State Convergence Tolerance  = 0
  Linear System Solver                = "Iterative"
  Linear System Preconditioning       = None
  Linear System Residual Output       = 0
  Linear System Max Iterations        = 5000
  Linear System Iterative Method      = "CG"
  Linear System Convergence Tolerance = 1.0e-8
  Linear System Symmetric             = True

  Nonlinear System Consistent Norm    = Logical True
  Discontinuous Bodies                = True
End


! ------------------------------------------------- !
! --- [4] Body Forces / Initial Conditions      --- !
! ------------------------------------------------- !

Body Force 1
  Name              = "CoilCurrentSource"
  Current Density 1 = Equals "CoilCurrent e 1"
  Current Density 2 = Equals "CoilCurrent e 2"
  Current Density 3 = Equals "CoilCurrent e 3"
End


! ------------------------------------------------- !
! --- [5] Boundary Conditions                   --- !
! ------------------------------------------------- !

Boundary Condition 1
  Name = "Far Boundary"
  Target Boundaries(1) = 201
  AV {e}               = 0.0
End

C型電磁石の入力ファイルの要点は以下である．

H-B Curveにより H-B 曲線をデータとして与えている．
Body 1 に付随する component を定義し、無次元量として理想電流 100 (kA) を与えている．
Body Force を定義し、componentが作っている電流を体積力として定義している．
Coil Closed を True としている．

円環コイル電流がつくる磁場の解析結果¶

解析実行結果は以下に示す．以下に電流密度分布と軸方向の磁束密度を示す．

../../../_images/cshape_magnet_bfield.png

C型電磁石がギャップ間につくる磁場は、アンペールの法則より、

$\int_C H \cdot dl &= \int_S J \cdot n dS \\ NI &= H_c l_c + H_g l_g = \dfrac{B}{\mu_c} l_c + \dfrac{B}{\mu_g} l_g \\ B &= \dfrac{ NI }{ \dfrac{l_c}{\mu_c} + \dfrac{l_g}{\mu_g} }$