Matching Interface Condition¶

The matching interface condition stitches two boundaries $\Gamma_0$ and $\Gamma_1$ together so that the magnetic vector potential $\mathbf{A}$ is continuous (or sign-flipped) across the interface.

The geometric matching maps side-1 onto side-0 by a rigid transform

\[ \mathbf{T}(\mathbf{x}) = \mathbf{R}\, \mathbf{x} + \mathbf{t}, \]

where $\mathbf{R}$ is an orthogonal $3 \times 3$ matrix (rotation or reflection) and $\mathbf{t}$ a translation vector. The orientation-aware HCurl coupling supports any orthogonal $\mathbf{R}$, so rotational sectors, periodic cells, and anti-periodic sectors can all be expressed directly.

Two symmetry modes are available:

Periodic (InterfaceSymmetry.PERIODIC) — the field is identical on both sides. Use for full-period sectors (one pole-pair of a motor, a full lattice cell, etc.).
Anti-periodic (InterfaceSymmetry.ANTI_PERIODIC) — the field reverses sign across the interface. Use for half-period sectors with alternating polarity (one pole pitch of a motor, a Halbach slab with alternating magnetization, etc.).

Periodic and anti-periodic matching of two interfaces

Figure 1: The transform $\mathbf{T}$ maps side-1 ($\Gamma_1$) onto side-0 ($\Gamma_0$). In the periodic mode the potential is copied across; in the anti-periodic mode it is sign-flipped.

Applicability¶

Applicable to a pair of boundary markers — one per side — that map onto each other under the rigid transform $\mathbf{T}$. The rotation $\mathbf{R}$ must be orthogonal. The condition couples the matched HCurl degrees of freedom (continuously for periodic, sign-flipped for anti-periodic); it does not model imperfect or partial coupling.

When to use this¶

Rotating machinery sector models. Solve one pole or one pole-pair of a synchronous machine, induction motor, or generator instead of the full $360°$ — typically a $4{-}10\times$ reduction in unknowns.
Periodic Halbach arrays and linear motors. A single sector with anti-periodic stitching reproduces an infinite alternating pattern.
Bulk lattices. Periodic unit cell of a structured material (e.g. a laminated stack viewed in plane) with translation-only matching.

Usage¶

A periodic boundary condition between two parallel faces separated by a translation vector:

from mufem.electromagnetics.timedomainmagnetic import (
    InterfaceSymmetry,
    MatchingInterfaceCondition,
)

periodic_bc = MatchingInterfaceCondition(
    name="Periodic",
    marker0="Cell::Left" @ Bnd,
    marker1="Cell::Right" @ Bnd,
    symmetry=InterfaceSymmetry.PERIODIC,
    translation=(-1.0, 0.0, 0.0),
)

time_domain_magnetic_model.add_condition(periodic_bc)

A $22.5°$ anti-periodic stitch (one sector of a 16-pole Halbach cylinder) around the $z$-axis:

import math

c = math.cos(math.pi / 8)
s = math.sin(math.pi / 8)

rotation_neg_22_5_z = [
    [ c,  s, 0.0],
    [-s,  c, 0.0],
    [0.0, 0.0, 1.0],
]

anti_periodic_bc = MatchingInterfaceCondition(
    name="AntiPeriodic",
    marker0="AntiPeriodic::Phi_0" @ Bnd,
    marker1="AntiPeriodic::Phi_22_5" @ Bnd,
    symmetry=InterfaceSymmetry.ANTI_PERIODIC,
    rotation=rotation_neg_22_5_z,
)

time_domain_magnetic_model.add_condition(anti_periodic_bc)

The rotation matrix must map vertices of marker1 into the frame of marker0. Translation defaults to no translation; rotation defaults to the identity.