Lorentz Force Coupling Analysis — A Method for Coupling Electromagnetic Forces and Structural Deformation

No, no, this is a topic structural engineers should definitely know about. For example, the superconducting magnet in an MRI—during excitation, the hundreds of amperes of current flowing through the coil experience Lorentz force in the strong magnetic field, trying to expand the winding. This force can reach several tons, so structural analysis must evaluate whether the support structure can withstand it.

A more familiar example is a transformer short-circuit fault. When the short-circuit current jumps to 10-25 times the normal level, an electromagnetic force of hundreds of kN is instantaneously applied between the coils. This can buckle the windings or damage the insulation. This is why power companies emphasize "transformer short-circuit withstand tests." Other examples include linear motors, electromagnetic forming, railguns—wherever there is current and a magnetic field, there is Lorentz force.

Exactly, the starting point is the force on a charged particle. The general form including both electric and magnetic fields is:

In coupled analysis for conductors in CAE, we use this extended to a continuum as a body force density. It's the cross product of current density $\mathbf{J}$ (A/m²) and magnetic flux density $\mathbf{B}$ (T):

Yes, like gravity, it enters the right-hand side of the FEM equations as a body force. Writing the structural equation of motion:

Here $\boldsymbol{\sigma}$ is the stress tensor, $\mathbf{f}_{\text{em}} = \mathbf{J} \times \mathbf{B}$ is the electromagnetic body force, $\rho$ is density, $\ddot{\mathbf{u}}$ is acceleration. For static analysis, the right side becomes 0. Roughly speaking, "solve for $\mathbf{J}$ and $\mathbf{B}$ in electromagnetic field analysis, take their cross product, and feed that force into structural analysis"—that's the skeleton of Lorentz force coupling.

Good question. J × B can only define force in regions where current flows. But magnetic materials like permanent magnets or ferrite cores experience force from the magnetic field even without current flow. That's when we use the Maxwell stress tensor:

Integrating this stress tensor over the object's surface gives the resultant electromagnetic force acting on the object:

Theoretically, it should give the same result for any closed surface surrounding the object, but numerically it can vary due to mesh coarseness. In practice, the "air gap integration" method, where the integration surface is placed slightly away from the object surface, is more stable. COMSOL uses this method by default.

Sharp observation. Magnetostriction is a different electromagnetic-structural coupling mechanism from Lorentz force. When a magnetic material is magnetized, the crystal lattice deforms minutely due to magnetic domain rotation. The transformer core repeatedly expands and contracts under the 50Hz/60Hz AC magnetic field, causing that humming sound.

Magnetostriction enters the structural equation as a strain tensor $\varepsilon^{\text{ms}}$:

The structure is the same as the thermal expansion formula $\sigma = C(\varepsilon - \alpha \Delta T)$, right? Magnetostrictive strain is typically on the order of $10^{-6}$ to $10^{-5}$, about $\lambda_s \approx 5 \times 10^{-6}$ for electrical steel sheets. It seems tiny, but when summed over the entire core, it becomes the main cause of transformer noise and vibration.

Exactly. In real problems, they often overlap. For example, in an electric motor, Lorentz force acts on the stator windings while magnetostrictive vibration occurs in the iron core. If you're doing noise analysis, you need to consider both to match experimental results.

Weak coupling is sufficient for many cases. Static evaluation of MRI magnets or short-circuit withstand analysis of transformers involve tiny structural deformations (on the order of mm or less), so their effect on the electromagnetic field is negligible.

However, there are cases where strong coupling is needed. A prime example is electromagnetic forming. This is a processing method where a high-speed deformation is induced in an aluminum sheet by instantaneously flowing eddy currents and their interaction with a magnetic field. If the distance between the coil and the workpiece changes due to deformation, the magnetic field and current distributions change significantly. One-way coupling would overestimate the force. The air gap variation problem in linear motors is similar.

For low-frequency electromagnetic field analysis, we use the quasi-static approximation that ignores displacement current from Maxwell's equations. The A-φ method formulated with magnetic vector potential $\mathbf{A}$ and scalar potential $\phi$ is standard:

$\mu$ is permeability, $\sigma$ is conductivity, $\mathbf{J}_s$ is externally applied current density. From this, we obtain magnetic flux density $\mathbf{B} = \nabla \times \mathbf{A}$, and the induced current density in the conductor is $\mathbf{J} = -\sigma(\partial \mathbf{A}/\partial t + \nabla \phi)$. Their cross product is the Lorentz force.

Exactly right. However, there's an important difference. In electromagnetic FEM, edge elements (Nedelec elements) are often used. With standard nodal elements, the tangential continuity of magnetic flux density isn't guaranteed. They're different from structural Lagrange elements, so care is needed when coupling because the element types differ.

There are mainly three methods. Each has strengths and weaknesses, so they are used according to the application:

The concept is the same. The force is obtained from the change in magnetic field energy when the object is given a virtual infinitesimal displacement $\delta s$:

Tools like JMAG or Ansys Maxwell automatically calculate this Virtual Work method as nodal forces. It offers a good balance of accuracy and stability, making it the de facto standard for torque calculations, etc.

This is the core of coupled analysis. There are three methods for mesh mapping:

It means the resultant force calculated on the electromagnetic side matches the resultant force after transfer to the structural side. If this breaks down, forces that don't exist can appear or disappear in the structural model. In transformer short-circuit force analysis, if the force conservation error exceeds 5%, the stress evaluation becomes quantitatively unusable, so always check the resultant force after transfer.

This is the tricky part of transient coupling. The electromagnetic field changes with skin effect time constants (milliseconds to microseconds), while the structural natural vibration period is on the order of milliseconds to seconds. It's common for the time scales to differ by 2-3 orders of magnitude.

Subcycling is an effective countermeasure. Solve the electromagnetic field with a fine time step $\Delta t_{\text{em}}$, and the structure with a coarse time step $\Delta t_{\text{str}} = n \cdot \Delta t_{\text{em}}$. For force transfer, at each structural step, average the electromagnetic force over that interval and pass it. For transformer short-circuit analysis, it's typical to run the electromagnetic field at $\Delta t = 0.1 \text{ms}$ and the structure at $\Delta t = 1 \text{ms}$.