Magnetism A-formulation

What you’ll learn

• Magnetostatic interactions — magnetic wall, remanence, and periodicity.
• Multiphysics couplings — A-v coupling to Current flow.
• Magnetic vector potential — when the A-formulation is the right choice for magnetostatics.
• Weak formulation — the variational form using magnetic vector potential $\boldsymbol{A}$.

This page describes the available interactions and couplings for magnetostatics using the magnetic vector potential $\boldsymbol{A}$ in Allsolve. The weak formulation is stated up front; the full derivation is at the end of this page.

Weak formulation

The weak form solved by Allsolve is

$$\begin{align} \int_{\Omega}\ \frac{1}{\mu}\ (\nabla \times \boldsymbol{A}) \cdot (\nabla \times \boldsymbol{A^\prime})\ d \Omega + \int_{\Omega}\ -\ \boldsymbol{J} \cdot \boldsymbol{A^\prime}\ d \Omega + \int_{\Gamma}\ (\boldsymbol{n} \times \boldsymbol{H}) \cdot \boldsymbol{A^\prime}\ d \Gamma = 0. \end{align}$$

where $\boldsymbol{A}$ is the magnetic vector potential, $\boldsymbol{H}$ the magnetic field intensity, $\boldsymbol{J}$ the current density, and $\mu$ the magnetic permeability.

See Formulation derivation for the full strong-to-weak derivation.

Interactions

Magnetic wall

Imposes a zero tangential magnetic field $\boldsymbol{n} \times \boldsymbol{H} = 0$ on the boundary, equivalent to a perfect magnetic conductor (PMC). In the A-formulation, this is a natural boundary condition, meaning it is satisfied automatically if no other condition is applied.

Remanence

Defines a remanent magnetization $\boldsymbol{B_r}$ 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, provide a 3×1 matrix for a 3D problem (2×1 for 2D, etc.) with remanence field vector values for each dimension. In the expression editor, write the direct expression.
Example	[1; 5; 0] applies a 3D remanence field of $B_{r,x} = 1\:T$, $B_{r,y} = 5\:T$ and $B_{r,z} = 0\:T$ to the specified region. Same principles apply for 2D. See also Adding spatially varying remanence field to permanent magnets.
Unit	Remanence magnetic field in Teslas (T)

Periodicity

Imposes periodic boundary conditions on the magnetic vector potential field $\boldsymbol{A}$ between two boundaries. Reduces the computational domain size for geometrically symmetric or antisymmetric problems, avoiding the need to model the full geometry.

Example	Periodicity follows the same principles, regardless of which physics module it belongs to. See how periodicity is used in magnetism as a reference: Periodicity in electric motors

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Couplings to Other Physics

This formulation supports the following couplings:

• A-v coupling
  • Current flow

Compatibilities with analysis types

Magnetism A supports Static, Harmonic, Multiharmonic, and Transient analysis. Eigenmode analysis is not supported.

See the full compatibility table in Simulations overview.

Use cases

• Magnetism A - Current flow coupling in a loudspeaker

Formulation derivation

Maxwell’s equations

$$\text{Gauss's law}$$$$\nabla \cdot \boldsymbol{D} = \rho$$$$\text{Gauss's law for magnetism}$$$$\nabla \cdot \boldsymbol{B} = 0$$$$\text{Faraday's law}$$$$\nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t}$$$$\text{Ampere-Maxwell law}$$$$\nabla \times \boldsymbol{H} = \boldsymbol{J} + \frac{\partial \boldsymbol{D}}{\partial t}$$

Strong formulation

Our starting points along with the material relation are

$$\begin{align*} \nabla \cdot \boldsymbol{B} &= 0 \tag{1} \\[5pt] \nabla \times \boldsymbol{H} &= \boldsymbol{J} \tag{2} \\[5pt] \boldsymbol{B} &= \mu \boldsymbol{H}. \tag{3} \end{align*}$$

Since the magnetic field is divergence-free, we can use the identity for divergence of a curl and write the magnetic field as a curl of some vector $\boldsymbol{A}$

$$\begin{align*} \boldsymbol{B} = \nabla \times \boldsymbol{A}. \tag{4} \end{align*}$$

Vector $\boldsymbol{A}$ is called a magnetic vector potential. By substituting $\text{(4)}$ into $\text{(3)}$ and rearranging, the material relation becomes

$$\begin{align*} \boldsymbol{H} = \frac{1}{\mu} \boldsymbol{B} = \frac{1}{\mu} \left( \nabla \times \boldsymbol{A} \right). \tag{5} \end{align*}$$

Substituting $\text{(5)}$ into $\text{(2)}$, the strong form becomes

$$\begin{align*} \nabla \times \boldsymbol{H} = \frac{1}{\mu} \left( \nabla \times \boldsymbol{A} \right) = \boldsymbol{J}. \tag{6} \end{align*}$$

Deriving the weak form

To obtain the weak form, multiply $\text{(2)}$ by the test function $\boldsymbol{A}^\prime$ and integrate over the domain $\Omega$:

$$\begin{align*} \int_{\Omega}\ (\nabla \times \boldsymbol{H} - \boldsymbol{J}) \cdot \boldsymbol{A^\prime}\ d \Omega = 0 \tag{7} \\[5pt] \int_{\Omega}\ (\nabla \times \boldsymbol{H}) \cdot \boldsymbol{A^\prime}\ d \Omega - \int_{\Omega}\ \boldsymbol{J} \cdot \boldsymbol{A^\prime}\ d \Omega = 0. \tag{8} \end{align*}$$

Using the identity for divergence of a cross product on the first term we get

$$\begin{align*} \int_{\Omega}\ \nabla \cdot (\boldsymbol{H} \times \boldsymbol{A^\prime})\ d \Omega + \int_{\Omega}\ \boldsymbol{H} \cdot (\nabla \times \boldsymbol{A^\prime})\ d \Omega - \int_{\Omega}\ \boldsymbol{J} \cdot \boldsymbol{A^\prime}\ d \Omega = 0. \tag{9} \end{align*}$$

Applying the divergence theorem on the first term we obtain

$$\begin{align*} \int_{\Gamma}\ (\boldsymbol{H} \times \boldsymbol{A^\prime}) \cdot \boldsymbol{n}\ d \Gamma + \int_{\Omega}\ \boldsymbol{H} \cdot (\nabla \times \boldsymbol{A^\prime})\ d \Omega - \int_{\Omega}\ \boldsymbol{J} \cdot \boldsymbol{A^\prime}\ d \Omega = 0. \tag{10} \end{align*}$$

Using the scalar triple product rule on the boundary term we get

$$\begin{align*} \int_{\Gamma}\ (\boldsymbol{n} \times \boldsymbol{H}) \cdot \boldsymbol{A^\prime}\ d \Gamma + \int_{\Omega}\ \boldsymbol{H} \cdot (\nabla \times \boldsymbol{A^\prime})\ d \Omega - \int_{\Omega}\ \boldsymbol{J} \cdot \boldsymbol{A^\prime}\ d \Omega = 0. \tag{11} \end{align*}$$

Substituting $\text{(5)}$ in the second term and by rearranging we get the final weak formulation stated at the top of this page.