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MagnetostaticPDEComponent

MagnetostaticPDEComponent[vars,pars]

yields a current-free magnetostatic PDE term with variables vars and pars.

Details

MagnetostaticPDEComponent is typically used to generate a magnetostatic equation for permanent magnets with model variables vars and model parameters pars.
MagnetostaticPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:

MagnetostaticPDEComponent models static magnetic fields produced by permanent magnets and other current-free magnetic sources.

MagnetostaticPDEComponent models magnetostatic phenomena with the dependent variable and the magnetic scalar potential. is in units of amperes [], independent variables in units of [].
Stationary variables vars are vars={V_m[x₁,…,x_n],{x₁,…,x_n}}.
MagnetostaticPDEComponent generally does not produces a time-dependent PDE.
To model permanent magnets, the MagnetostaticPDEComponent equation is given as:

is the vacuum permeability in units of [] and the magnetization vector in units of [].
The magnetization vector specifies the magnetic dipole moment per unit volume within a material, indicating the strength and direction of its magnetic properties.
An alternative to the magnetization vector , is the remanent magnetic flux density vector in units of [ $TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]. The MagnetostaticPDEComponent equation is given as:

is the unitless recoil permeability.
For linear materials, like iron, the MagnetostaticPDEComponent equation simplifies to:

is the unitless relative permeability.
can be isotropic, orthotropic or anisotropic.
can be a function of the magnetic field and describe nonlinear materials.
The implicit default boundary condition for the magnetostatic model is a 0 MagneticFluxDensityValue.
The units of the magnetostatic model terms are in [ $TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 3}}, webers per meter cubed, {{(, "Webers", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ].
The following parameters pars can be given:

parameter	default	symbol
"Magnetization"	{0,…}	, magnetization vector in []
"RegionSymmetry"	None
"RelativePermeability"	1	, unitless relative permeability
"RemanentMagneticFluxDensity"	{0,…}	, remanent magnetic flux density in [ $TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]
"Thickness"	1	, thickness in []
"VacuumPermeability"		, vacuum permeability in []

All parameters may depend on the spatial variable and dependent variable .
The number of independent variables determines the dimensions of or and the length of vectors , and .
The models are available in a 2D, a 2D axisymmetric and a 3D form.
A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
"Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
dimension reduction equation

2D
In 2D, when a "Thickness" is specified, the MagnetostaticPDEComponent equation is given as:

The input specification for the parameters is exactly the same as for their corresponding operator terms.
If no parameters are specified, the default magnetostatic PDE is:

If the MagnetostaticPDEComponent depends on parameters that are specified in the association pars as …,keyp_i…,p_iv_i,…, the parameters are replaced with .

Examples

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Basic Examples (3)

Define a default magnetostatic PDE model:

Wolfram Language code: MagnetostaticPDEComponent[{Subscript[V, m][x, y], {x, y}}, <||>]

Set up a magnetostatic model with particular material parameters:

Wolfram Language code: MagnetostaticPDEComponent[{Subscript[V, m][x, y, z], {x, y, z}}, <|"RelativePermeability" -> Subscript[μ, 𝓇]|>]

To model a permanent magnet in 2D with a rectangular cross section, define the mesh to use:

Wolfram Language code: mesh = \!$\*GraphicsBox[«2»]$;

Visualize the internal boundaries of the magnet region:

Wolfram Language code: Show[HighlightMesh[mesh, {...}], PlotRange -> {{-0.08, 0.08}, {-0.15, 0.15}}]

Solve the magnetostatic PDE model with a magnet transversely magnetized in the direction of the axis:

Wolfram Language code:

VmFun = NDSolveValue[{MagnetostaticPDEComponent[{Vm[x, y], {x, y}}, <|"Magnetization" -> {0, Piecewise[{{400000, RegionMember[Rectangle[{-0.05, -0.1}, {0.05, 0.1}]][{x, y}]}}, 0]}|>] == 0, MagneticPotentialCondition[x == -1 || x == 1 || y == -1 || y == 1, {Vm[x, y], {x, y}}, <||>]}, Vm, {x, y}∈mesh]

Visualize the magnetic field:

Wolfram Language code:

Show[Graphics[{Gray, Rectangle[{-0.05, -0.1}, {0.05, 0.1}]}], StreamPlot[Evaluate[-Grad[VmFun[x, y], {x, y}]], {x, -0.3, 0.3}, {y, -0.3, 0.3}, StreamColorFunction -> "Rainbow"]]

Scope (3)

Specify a magnetostatic PDE with a magnetization vector of in units of []:

Wolfram Language code:

MagnetostaticPDEComponent[{Subscript[V, m][x, y], {x, y}}, <|"Magnetization" -> {0, Quantity[200000, "Amperes"/"Meters"]}|>]

Define a symbolic 2D axisymmetric magnetostatic PDE:

Wolfram Language code:

MagnetostaticPDEComponent[{Subscript[V, m][r, z], {r, z}}, <|"RelativePermeability" -> Subscript[μ, 𝓇], "VacuumPermeability" -> Subscript[μ, 0], "RegionSymmetry" -> "Axisymmetric"|>]

Define a 3D magnetostatic model with particular material parameters:

Wolfram Language code:

MagnetostaticPDEComponent[{Subscript[V, m][x, y, z], {x, y, z}}, <|"RelativePermeability" -> Subscript[μ, rec], "VacuumPermeability" -> Subscript[μ, 0], "RemanentMagneticFluxDensity" -> {Brx[x, y, z], Bry[x, y, z], Brz[x, y, z]}|>]

Applications (1)

To model a 3D cylinder permanent magnet, set up variables:

Wolfram Language code: vars = {Vm[x, y, z], {x, y, z}};

Define the magnet region of height [] and radius []:

Wolfram Language code: magnet = Cylinder[{{0, 0, -1 / 10}, {0, 0, 1 / 10}}, 1 / 10];

Define the magnetization vector:

Wolfram Language code: pars = <|"Magnetization" -> {0, 0, Piecewise[{{1*^5, RegionMember[magnet][{x, y, z}]}}, 0]}|>;

Set up boundary conditions:

Wolfram Language code: Subscript[Γ, p] = MagneticPotentialCondition[x ^ 2 + y ^ 2 + z ^ 2 >= 2 ^ 2, vars, pars]

Set up the mesh with a sphere of air of [] that represents the surrounding volume:

Wolfram Language code: mesh = \!$\*Graphics3DBox[«5»]$;

Visualize the magnet cylinder that is inside the mesh:

Wolfram Language code: Show[Graphics3D[{Opacity[0.1], magnet}, Boxed -> False], HighlightMesh[RegionBoundary[mesh], {}, PlotTheme -> "Lines"]]

Set up the magnetostatIc PDE component:

Wolfram Language code: op = MagnetostaticPDEComponent[vars, pars];

Solve the PDE:

Wolfram Language code: VmFun = NDSolveValue[{op == 0, Subscript[Γ, p]}, Vm, {x, y, z}∈mesh]

Visualize the magnetic field:

Wolfram Language code:

Show[Graphics3D[{LightGray, magnet}], StreamPlot3D[Evaluate[-Grad[VmFun[x, y, z], {x, y, z}]], {x, -0.5, 0.5}, {y, -0.5, 0.5}, {z, -0.5, 0.5}, ...], Rule[...]]

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MagnetostaticPDEComponent

Details

Examples

Basic Examples (3)

Scope (3)

Applications (1)

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	dimension	reduction	equation
	2D

MagnetostaticPDEComponent

Details

Examples

Basic Examples (3)

Scope (3)

Applications (1)

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Text

CMS

APA

BibTeX

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