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Magnetism 𝜑

The Magnetism φ formulation solves for the magnetic scalar potential to describe magnetostatic fields. This approach is particularly efficient for large 3D non-conducting domains, where using the magnetic vector potential A would lead to a significantly higher number of degrees of freedom (DOFs).

Strong formulation

Section titled “Strong formulation”

Our starting points along with the material relation are

∇⋅B=0∇×H=JB=μH.\begin{align} \nabla \cdot \boldsymbol{B} &= 0 \\[5pt] \nabla \times \boldsymbol{H} &= \boldsymbol{J} \\[5pt] \boldsymbol{B} &= \mu\boldsymbol{H}. \end{align}

Under the assumption of no current flow (J=0\boldsymbol{J} = 0), Gauss’s law for magnetism and Ampere’s law state

∇⋅B=0∇×H=0.\begin{align} \nabla \cdot \boldsymbol{B} &= 0 \\[5pt] \nabla \times \boldsymbol{H} &= 0. \end{align}

The curl of H\boldsymbol{H} is zero, so H\boldsymbol{H} is a conservative field, meaning there exists a scalar function φ\varphi such that

H=−∇φ.\begin{align} \boldsymbol{H} = -\nabla \varphi. \end{align}

Here, φ\varphi is called the magnetic scalar potential. Substituting (5) and the constitutive relation B=μH\boldsymbol{B} = \mu \boldsymbol{H} into Gauss’s law for magnetism we get

∇⋅μH=0∇⋅μ∇φ=0,\begin{align} \nabla \cdot \mu \boldsymbol{H} &= 0 \\[5pt] \nabla \cdot \mu \nabla \varphi &= 0, \end{align}

resulting in Laplace’s equation.

Weak formulation

Section titled “Weak formulation”

The partial differential equation (8)(8) is multiplied by the test function φ′\varphi^{\prime} and integrated over the entire domain Ω\Omega to get

∫Ω  ∇⋅(μ∇φ)  φ′  dΩ=0.\begin{align} \int_{\Omega}\; \nabla \cdot (\mu \nabla \varphi)\; \varphi^{\prime}\; d \Omega = 0. \end{align}

Using the Leibniz rule for nabla operator on the divergence term we get

∫Ω  ∇⋅(φ′  μ∇φ)  dΩ+∫Ω  −(∇φ′⋅μ∇φ)  dΩ=0.\begin{align} \int_{\Omega}\; \nabla \cdot (\varphi^{\prime}\; \mu \nabla \varphi)\; d \Omega + \int_{\Omega}\; -(\nabla \varphi^{\prime} \cdot \mu \nabla \varphi)\; d \Omega = 0. \end{align}

Applying the divergence theorem on the divergence term we get

∫Γ  μ∇φ⋅n φ′  dΓ+∫Ω  −(μ∇φ⋅∇φ′)  dΩ=0.\begin{align} \int_{\Gamma}\; \mu \nabla \varphi \cdot \boldsymbol{n}\ \varphi^{\prime}\; d \Gamma + \int_{\Omega}\; -(\mu\nabla \varphi \cdot \nabla \varphi^{\prime})\; d \Omega = 0. \end{align}

Rearranging the terms and using relation B=μH=−μ∇φ\boldsymbol{B} = \mu \boldsymbol{H} = -\mu \nabla \varphi on the Neumann term, we obtain the final weak formulation

∫Ω  (−μ∇φ⋅∇φ′)  dΩ+∫Γ  (−B⋅n)  φ′  dΓ=0.\begin{align} \int_{\Omega}\; (-\mu\nabla \varphi \cdot \nabla \varphi^{\prime})\; d \Omega + \int_{\Gamma}\; (-\boldsymbol{B} \cdot \boldsymbol{n}) \; \varphi^{\prime}\; d \Gamma = 0. \end{align}

Available Interactions

Section titled “Available Interactions”

Constraint

Section titled “Constraint”

Applies a fixed value to the scalar potential 𝜑. Use this when you need to fix scalar potential at a node or within a region. This is often used as a gauge condition to ensure a unique solution.

How to use:

Provide a scalar field value in point or region in Amperes

Example:

𝜑𝜑 = 1000 applies a magnetic scalar potential of 1000 A to the specified region.

Unit: Magnetic scalar potential in Amperes (A)

Remanence

Section titled “Remanence”

Defines a remanent magnetization Br\boldsymbol{B_r} for permanent magnets. Can be specified as a constant vector or as a vector valued function.

How to use:

You can use either the matrix editor or the expression editor.

With the matrix editor, you must provide a 3×1 matrix for a 3D problem (2×1 for 2D, etc.):

Add remanence field vector values for each dimension, for example Br,x=1B_{r,x} = 1, Br,y=5B_{r,y} = 5 and Br,z=0B_{r,z} = 0 for 3D case.

In the expression editor, you can write the direct expression for this 3D case:

Example: [1; 5; 0]

This applies a 3D remanence field constraint of Br,x=1 TB_{r,x} = 1\:T, Br,y=5 TB_{r,y} = 5\:T and Br,z=0 TB_{r,z} = 0\:T to specified node or a region. Same principles apply for 2D.

Adding spatially varying remanence field to permanent magnets

Unit: Remanence magnetic field in Teslas (T)

Lump φ/Φ

Section titled “Lump φ/Φ”

Applies a lumped magnetic scalar potential 𝜑𝜑 or magnetic flux ΦΦ to a curve in the domain. Used to drive magnetomotive force or magnetic flux through a specific path, such as along a coil, without resolving the full field distribution.

How to use:

Specify the target curve. From Actuation mode, select either magnetic scalar potential, magnetic flux or circuit coupling. Fill in the value.

Example: 𝜑=1000𝜑 = 1000 applies a magnetic scalar potential of 1000 A to the specified curve.

Unit: Magnetic scalar potential in Amperes (A) or magnetic flux in Webers (Wb)

Lump I/V Cut

Section titled “Lump I/V Cut”

Applies a lumped current II or voltage VV through a cut surface in the domain. In the magnetic scalar potential formulation, cuts are required to make the domain simply connected, ensuring a unique solution for 𝜑. This interaction is used to drive current or voltage through conducting regions such as coils, without explicitly resolving the detailed field distribution inside the conductor.

How to use:

Specify the target curve. It has to be closed boundary of some surface. From Actuation mode, select either current, voltage or circuit coupling. Fill in the value.

Example: I=100I = 100 applies 100 Amperes through the cut surface.

Lump I/V Cut in Induction motor

Unit: Current in Amperes (A) or voltage in Volts (V)

External Field

Section titled “External Field”

Applies an external magnetic flux density Bext\boldsymbol{B}_{ext} to the simulation domain. This represents an imposed background field that interacts with the model. The field can be specified as a constant vector or spatially varying field.

How to use:

You can use either the matrix editor or the expression editor.

With the matrix editor, you must provide a 3×1 matrix for a 3D problem (2×1 for 2D, etc.). Add external magnetic field values for each dimension.

In the expression editor, you can write the direct expression:

Example: [1; 5; 0]

applies a 3D external magnetic field constraint of Bext,x=1 TB_{ext,x} = 1\:T, Bext,y=5 TB_{ext,y} = 5\:T and Bext,z=0 TB_{ext,z} = 0\:T to specified node or a region. Same principles apply for 2D.

Unit: External magnetic field in Teslas (T)

Periodicity

Section titled “Periodicity”

Imposes periodic boundary conditions on the magnetic scalar potential 𝜑𝜑 between two boundaries. Reduces the computational domain size for geometrically symmetric or antisymmetric problems, avoiding the need to model the full geometry.

Example:

The periodicity of an electric motor allows modeling only a fraction of the full geometry, such as one pole pair or one quarter, while still capturing the complete field behavior.

Periodicity in electric motor

Continuity

Section titled “Continuity”

Ensures continuity of the magnetic scalar potential φ\varphi across an interface between two boundaries. Used to map the magnetic field continuously.

Example:

For example, continuity maps the rotor and stator fields correctly in rotating mesh interfaces in electric motors.

Continuity in electric motor

Use cases

Section titled “Use cases”