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Commit 89d31f4b authored by François Henrotte's avatar François Henrotte
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...@@ -2,7 +2,8 @@ ...@@ -2,7 +2,8 @@
Tutorial 9 : 3D magnetostatic dual formulations and magnetic forces Tutorial 9 : 3D magnetostatic dual formulations and magnetic forces
Features: Features:
- Dual 3D magnetostatic formulations - 3D Magnetostatics
- Dual vector and scalar magnetic potentials formulations
- Boundary condition at infinity with infinite elements - Boundary condition at infinity with infinite elements
- Maxwell stress tensor and rigid-body magnetic forces - Maxwell stress tensor and rigid-body magnetic forces
...@@ -24,12 +25,12 @@ ...@@ -24,12 +25,12 @@
in the problem decription below, irresective of whether they are in the problem decription below, irresective of whether they are
truly permanent magnets or ferromagnetic barrels. truly permanent magnets or ferromagnetic barrels.
The tutorial model proposes the both dual 3D magnetostatic formulations: The tutorial model proposes both dual 3D magnetostatic formulations:
the magnetic vector potential formulation with spanning-tree gauging, the magnetic vector potential formulation with spanning-tree gauging,
and the scalar magnetic potential formulation. and the scalar magnetic potential formulation.
The later is rather simple in this case since, as there are no conductors, The later is rather simple in this case since, as there are no conductors,
the known coercive field hc[] is the only source field "hs" one needs the known coercive field hc[] is the only source field "hs" one needs
to represens the magnetic field: to represent the magnetic field:
h = hs - grad phi , hs = -hc. h = hs - grad phi , hs = -hc.
...@@ -45,19 +46,22 @@ ...@@ -45,19 +46,22 @@
which is a material dependent function of the magnetic induction "b" field. which is a material dependent function of the magnetic induction "b" field.
Exactly like we computed the heat flux "q(S)" through a surface "S" Exactly like we computed the heat flux "q(S)" through a surface "S"
using a special auxiliary function "g(S)" associated with that surface using a special auxiliary function "g(S)" associated with that surface
in the tutorial "Tutorial 5 : thermal problem with contact resistances", in "Tutorial 5 : thermal problem with contact resistances",
the magnetic force acting on a rigid body in empty space can be evaluated the magnetic force acting on a rigid body in empty space can be evaluated
as the flux of the Maxwell stress tensor through that surface. as the flux of the Maxwell stress tensor through that surface.
There is one auxiliary function "g(SkinMagnet~{i}) = un~{i}" There is one auxiliary function "g(SkinMagnet~{i}) = un~{i}"
for each magnet and the resultant magnetic force acting on "Magnet~{i}" for each magnet, and the resultant magnetic force acting on "Magnet~{i}"
is given by the integral: is given by the integral:
f~{i} = Integral [ TM[{b}] * {-grad un~{i}} ] ; f~{i} = Integral [ TM[{b}] * {-grad un~{i}} ] ;
It should be insisted on the fact that the Maxwell stress is discontinuous It should be insisted on the fact that the Maxwell stress tensor
on material discontinuities, and that magnetic forces on rigid bodies is always discontinuous on material discontinuities,
only depend on the Maxwell stress tensor of empty space and and that magnetic forces acting on rigid bodies
on the "b" and "h" field distribution on the outer side of "SkinMagnet~{i}". only depend on the Maxwell stress tensor of empty space,
and on the "b" and "h" field distribution,
on the external side of "SkinMagnet~{i}"
(the side of the surface in contact with air).
"{-grad un~{i}}" in the above formula can be regarded "{-grad un~{i}}" in the above formula can be regarded
as the normal vector to "SkinMagnet~{i}" as the normal vector to "SkinMagnet~{i}"
...@@ -69,7 +73,7 @@ ...@@ -69,7 +73,7 @@
which is much smaller than "AirBox". which is much smaller than "AirBox".
To speed up the computation of forces, a special domain "Vol_Force" To speed up the computation of forces, a special domain "Vol_Force"
for force integrations is defined, which contains only for force integrations is defined, which contains only
the layers "layer~{i}" of alla magnets. the layers "layer~{i}" of all magnets.
*/ */
Include "magnets_common.pro" Include "magnets_common.pro"
......
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