Magnetism H

Strong formulation

The $\boldsymbol{H}$ -formulation is derived based on the magnetostatic approximation. Our starting equations are Faraday’s law and Ampere’s law.

\begin{align} \nabla \times \boldsymbol{E} &= -\frac{\partial \boldsymbol{B}}{\partial t} \\[10pt] \nabla \times \boldsymbol{H} &= \boldsymbol{J} \end{align}

Using Ohm’s law $\boldsymbol{J} = \sigma \boldsymbol{E}$ and plugging in the Ampere’s law we get

\begin{align} \boldsymbol{E} = \frac{1}{\sigma} (\nabla \times \boldsymbol{H})\ . \end{align}

Using the material relation $\boldsymbol{B} = \mu \boldsymbol{H}$ and substituting it into Faraday’s law we end up with

\begin{align} \nabla \times \boldsymbol{E} = -\frac{\partial \mu \boldsymbol{H}}{\partial t}\ . \end{align}

Substituting equation (3) into equation (4) and rearranging we derive the $\boldsymbol{H}$ -formulation

\begin{align} \nabla \times (\frac{1}{\sigma} \nabla \times \boldsymbol{H}) + \frac{\partial (\mu \boldsymbol{H})}{\partial t} = 0 \ . \end{align}

Weak formulation

The partial differential equation (4) is multiplied by the test vector $\boldsymbol{H}^{\prime}$ and integrated over the entire domain $\Omega$ .

\begin{align} \int_{\Omega}\ (\nabla \times \boldsymbol{E} + \frac{\partial (\mu \boldsymbol{H})}{\partial t}) \cdot \boldsymbol{H}^{\prime}\ d \Omega &= 0 \\[10pt] \int_{\Omega}\ ( \nabla \times \boldsymbol{E}) \cdot \boldsymbol{H}^{\prime}\ d \Omega + \int_{\Omega}\ \frac{\partial (\mu \boldsymbol{H})}{\partial t} \cdot \boldsymbol{H}^{\prime}\ d \Omega &= 0 \ . \end{align}

Applying the divergence of a cross product on the first term we have

\begin{align} \int_{\Omega}\ \nabla \cdot (\boldsymbol{E} \times \boldsymbol{H}^{\prime})\ d\Omega + \int_{\Omega}\ \boldsymbol{E} \cdot (\nabla \times \boldsymbol{H}^{\prime})\ d\Omega + \int_{\Omega}\ \frac{\partial (\mu \boldsymbol{H})}{\partial t} \cdot \boldsymbol{H}^{\prime}\ d\Omega = 0 \ . \end{align}

Using the divergence theorem on the divergence term we obtain

\begin{align} \int_{\Gamma}\ (\boldsymbol{E} \times \boldsymbol{H}^{\prime}) \cdot \boldsymbol{n}\ d\Gamma + \int_{\Omega}\ \boldsymbol{E} \cdot (\nabla \times \boldsymbol{H}^{\prime})\ d\Omega + \int_{\Omega} \frac{\partial (\mu \boldsymbol{H})}{\partial t} \cdot \boldsymbol{H}^{\prime}\ d\Omega = 0 \ . \end{align}

Making use of the scalar triple product rule on the boundary and rearranging we get

\begin{align} \int_{\Omega}\ \boldsymbol{E} \cdot (\nabla \times \boldsymbol{H}^{\prime})\ d\Omega + \int_{\Omega} \frac{\partial (\mu \boldsymbol{H})}{\partial t} \cdot \boldsymbol{H}^{\prime}\ d\Omega + \int_{\Gamma} (\boldsymbol{n} \times \boldsymbol{E}) \cdot \boldsymbol{H}^{\prime}\ d\Gamma = 0 \ , \end{align}

and by substituting equation (3) into the first term we derive the weak formulation

\begin{align} \int_{\Omega} \frac{1}{\sigma} (\nabla \times \boldsymbol{H}) \cdot (\nabla \times \boldsymbol{H}^{\prime})\ d\Omega + \int_{\Omega} \frac{\partial (\mu \boldsymbol{H})}{\partial t} \cdot \boldsymbol{H}^{\prime}\ d\Omega + \int_{\Gamma} (\boldsymbol{n} \times \boldsymbol{E}) \cdot \boldsymbol{H}^{\prime}\ d\Gamma = 0 \ . \end{align}

Available Interactions

Constraint

Applies a fixed value to the $\boldsymbol{H}$ field. Use this when you need to fix $\boldsymbol{H}$ at a node or within a region.

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 magnetic field strength vector values for each dimension.

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

Example: [1; 5; 0]

Both styles are equivalent and apply a 3D magnetic field strength constraint of $H_x = 1\:\frac{A}{m}$ , $H_y = 5\:\frac{A}{m}$ and $H_z = 0\:\frac{A}{m}$ to specified node or a region. Same principles apply for 2D.

Unit: Magnetic field strength in Amperes per meter (A/m)

Periodicity

Imposes periodic boundary conditions for the $\boldsymbol{H}$ . Reduces computational domain size for symmetric problems.

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

Example:

Periodicity is similar in every physics section. This example is from $\varphi$ -formulation, but workflows are identical:

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

Couplings to Other Physics

This formulation supports the following couplings:

H-𝜑 coupling (Magnetism 𝜑)
The H–𝜑 coupling enables a mixed formulation, where one part of the domain uses the magnetic field intensity H as the primary variable, and another part uses the magnetic scalar potential 𝜑. This approach is particularly useful for models that combine non-conducting and conducting regions, allowing each region to use the most efficient formulation.

Electrical insulator (Magnetism 𝜑)