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

Our starting points along with the material relation are

\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 ( $\boldsymbol{J} = 0$ ), Gauss’s law for magnetism and Ampere’s law state

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

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

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

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

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

resulting in Laplace’s equation.

Weak formulation

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

\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

\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

\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 $\boldsymbol{B} = \mu \boldsymbol{H} = -\mu \nabla \varphi$ on the Neumann term, we obtain the final weak formulation

\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

Constraint

Applies a fixed value to the scalar potential. 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.

Remanence

Defines a remanent magnetization for permanent magnets. Can be specified as a constant value or as a vector field.

Lump φ/Φ

Applies a lumped scalar potential 𝜑 or magnetic scalar potential Φ to a specific region or node. This is often used to model simplified circuit elements, where the detailed field distribution is not explicitly resolved but replaced with an equivalent lumped potential difference.

Lump I/V Cut

Applies current/voltage cuts through the domain. Used to drive current or voltage trough regions.

External Field

Applies an external magnetic flux density B 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.

Periodicity

Imposes periodic boundary conditions on 𝜑. Reduces computational domain size for symmetric problems.

Continuity

Ensures continuity of scalar potential 𝜑 across material interfaces. Used to map the fields, for example in rotating simulations.