Theory Reference
Magnetic Fields Calculations
The primary purpose of this library is to compute the magnetic fields generated by a source domain and evaluated at a target domain. The source domain is generally assumed to be unaffected by the response of the target domain.
Biot-Savart Law
The Biot-Savart Law (Ref 1) underpins the majority of calculations: $$ B(\vec{r}) = \frac{\mu_0}{4\pi} \iiint_V \frac{(\vec{J}dV) \times \vec{r'}}{|\vec{r'}|^3} $$
A closed-form solution to this volume integral does not exist for the vast majority of geometries. Instead, this library operates on geometries that have been discretized into finite elements. However, finite element theory is generally not employed in the solvers.
Point Sources
oersted's solvers act on two types of sources: "point" representations of elements,
and 4-node (linear) tetrahedra. Point representations of an element are developed by
treating the Biot-Savart law as a discrete sum instead of an exact integral:
In this formulation, the solution is assumed to converge when the number of elements in the mesh extends to infinity. In practice, this offers a high-quality representation of the far-field (several element lengths between source and target), but average errors with the mesh are much higher. However, the force calculation has a smoothing (integrating) effect and resultant Lorentz force calculations typically converge easily.
Current Density Source
Magnetic fields generated by point sources carrying electrical currents are computed according to: $$ B(\vec{r}) = \frac{\mu_0}{4\pi} \frac{(\vec{J_i}\cdot v_i) \times \vec{r_i'}}{|\vec{r_i'}|^3} $$
Magnetized Material (Dipole) Source
Magnetic fields generated by a point source with an associated magnetic moment are computed according to: $$ H(r) = \frac{1}{4\pi} (\frac{3r(m\cdot r)}{|r|^5} - \frac{m}{|r|^3}) $$
Finite Element Sources
Finite element sources are treated by computing the analytic Biot-Savart integral over the element. This means that the solution is exact, save for the following sources of error:
- When the edges of the element fail to adequately capture the surface of a geometry (e.g. a coarse mesh of a curved surface)
- When the mesh discretization fails to properly resolve the current density distribution with a conductor
Otherwise, the calculation is numerically accurate to floating point precision.
Notes:
- Finite element integration is extremely computationally costly; use of the octree (hybrid) solver is recommended for meshes > 100K elements
This library follows the procedures described in Ref 5 & 6. First, the volume integral is decomposed into a summation of surface integrals. Each of these surface integrals is then decomposed into a summation of line integrals.
Force Calculation
Forces on magnetic materials and current-carrying conductors are calculated according to the following equations:
Lorentz Force
Notes:
- Lorentz forces are volumetric and therefore calculated on a per-element basis within the mesh
Maxwell Force
The Maxwell stress tensor is defined as:
and can be integrated over a surface to find the total force acting on a body: $$ F = \int\int{S\cdot \hat{n} \space dA }$$
Notes:
oersted's implementation of the Maxwell stress tensor is mesh-sensitive and requires a refinement study. Be careful with the results.- The Maxwell stress tensor is the most general-purpose of the force outputs and can be used for both magnetized material and current-carrying conductors
Kelvin Force
The Kelvin force is defined as:
Where:
- \(m = M \cdot V\) is the magnetic moment
- \(B\) is the magnetic flux density from all background sources, excluding the body itself
- \(\nabla\) is the gradient operator
Notes:
- The Kelvin force is much less mesh-sensitive than the Maxwell stress tensor
- Fields should be computed on the body as if the body itself was not magnetized
- The Kelvin force cannot compute forces on current-carrying conductors with unity magnetic permeability.