Boundary conditions¶

Problem statement¶

Boundary conditions close the variational or finite-difference problem at every physical and auxiliary-field boundary. FullMag keeps the physical boundary law, discretisation realization, selected backend, and resolved boundary markers distinct. A periodic magnetization seam, a natural exchange boundary, a DMI surface law, and an FEM airbox closure are different contracts.

Governing equations¶

The free exchange surface law, periodic identification, and airbox closure are represented by

The DMI term is present only for the corresponding interfacial realization and \(\beta\) is the resolved Robin coefficient. A Dirichlet airbox closure \(u=0\) is a different discrete problem.

Boundary conditions define how the magnetization and auxiliary fields (scalar potential, displacement) behave at the edges of the computational domain. FullMag implements several boundary-condition types, each owned by the relevant interaction or solver module.

Free-surface Neumann (natural exchange boundary)¶

The default magnetic boundary condition in bulk micromagnetics is the homogeneous Neumann condition on the reduced magnetization:

where \(\mathbf{n}\) is the outward unit normal. This represents zero exchange torque at free surfaces — the magnetization is free to rotate without constraint at the boundary.

In FEM implementations, this condition is the natural (variational) boundary condition and requires no explicit enforcement. In FDM implementations, open or inactive neighbours are replaced by the centre magnetization, yielding zero normal flux.

DMI-modified boundary conditions¶

When Dzyaloshinskii–Moriya interaction (DMI) is active, the natural boundary condition on \(\partial\Omega_m\) is modified. The exchange-plus-DMI surface term becomes

or the corresponding bulk DMI form. This boundary condition arises naturally from the variational principle when the DMI weak form is included. FullMag enforces it through the FEM weak formulation; FDM implementations handle it through stencil-boundary modifications.

See Dzyaloshinskii–Moriya interaction for the full DMI boundary-condition documentation.

Periodic boundary conditions¶

Periodic boundary conditions (PBC) impose translational symmetry along one or more Cartesian axes. For the magnetization:

where \(L_d\) is the period length along direction \(d\).

# %% Periodic boundary request through the stage-first study API
import fullmag as fm

nm = 1.0e-9
study = fm.study("periodic-boundary-example")
study.engine("fem")
study.device("cpu", precision="double")
study.mode("strict")
study.objects.mesh.defaults(cell_size=(2 * nm, 2 * nm, 2 * nm))
study.pbc(x=True, y=True, demag="open")
study.exchange()
body = study.geometry(fm.Box(40 * nm, 40 * nm, 2 * nm), name="film")
body.Ms = 800.0e3
body.Aex = 13.0e-12
body.alpha = 0.01
body.m = fm.init.UniformMagnetization((1.0, 0.0, 0.0))
study.stages.add_relax(stage_id="relax", dt=1.0e-15, max_steps=10, tolT=1.0e-6)
study.stages.add_run(stage_id="run", until=1.0e-12)

Airbox boundary conditions (demagnetization)¶

The FEM Poisson-demag solver operates on a domain larger than the magnetic body. The airbox boundary supports two modes:

1. Dirichlet: \(u = 0\) on \(\partial\Omega_{\mathrm{air}}\). Accuracy improves with airbox size, but large airboxes increase computational cost.

2. Robin (asymptotic): \(\partial_n u + u/r = 0\) on \(\partial\Omega_{\mathrm{air}}\). This first-order asymptotic condition yields better accuracy at smaller airbox scales.

See Boundary Conditions for the full demagnetization boundary-condition documentation.

Mechanical boundary conditions¶

For magnetoelastic simulations, the elastic displacement field \(\mathbf{u}\) requires its own boundary conditions:

• Free surface (traction-free): \(\boldsymbol{\sigma}\cdot\mathbf{n} = \mathbf{0}\)

• Clamped (fixed): \(\mathbf{u} = \mathbf{0}\)

• Prescribed traction: \(\boldsymbol{\sigma}\cdot\mathbf{n} = \mathbf{t}\)

See Magnetoelastic interaction for details.

Symbols and SI units¶

Scientific bibliography¶

1. W. F. Brown Jr., Micromagnetics, Interscience Publishers, New York, 1963.

2. S. Rohart and A. Thiaville, “Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction,” Physical Review B 88, 184422 (2013). doi:10.1103/PhysRevB.88.184422.

3. D. R. Fredkin and T. R. Koehler, “Hybrid method for computing demagnetizing fields,” IEEE Trans. Magn. 26(2), 415 (1990).

Assumptions and validity¶

• Normals are outward normals of the resolved physical boundary marker, not of an arbitrary mesh face selected after meshing.

• PBC pairs must be complete, orientation-consistent, and compatible with the selected mesh.

• Natural FEM conditions arise from the weak form; they are not interchangeable with strong nodal clamps.

• Dirichlet and Robin demagnetization closures require separate convergence studies.

Python API¶

The stage-first declaration records periodic axes and the demagnetization boundary policy. The interaction-specific pages own the detailed parameter contracts.

Canonical ProblemIR¶

{
  "pbc": {
    "axes": ["periodic", "periodic", "open"],
    "demag": "open"
  }
}

The request is distinct from resolved periodic node pairs, boundary markers, gauge constraints, and the backend-specific operator. Those are recorded in the resolved plan and provenance.

Round-trip and failure semantics¶

Requested intent includes axis policy, demag closure, and mechanical boundary choice. Resolved execution includes paired nodes/faces, operator reduction, boundary marker, solver, device, and precision. Validation errors cover malformed axes, incomplete pairs, incompatible periodic mesh topology, and conflicting strong/natural conditions. Unsupported combinations fail closed; no periodicity or boundary law is silently dropped.

Discrete realization¶

Implementation mapping¶

Python owns the requested policy, the planner resolves legality and pairs, FDM owns wrapped neighbors, and FEM owns weak-form/constraint realization. Demagnetization-specific closures remain under the demagnetization subtree.

Validation¶

Validate pair completeness and residuals, zero exchange flux on free surfaces, DMI boundary variation, periodic seam equality, gauge/airbox convergence, FEM reduced-system consistency, FDM CPU/GPU parity, and CPU/GPU device identity. Compare Dirichlet and Robin demag only as separate qualification cases.

Limitations¶

Not every combination of periodicity, FEM airbox, DMI, and mechanical constraints is executable. The planner status is authoritative for a concrete request; this shared page does not promote a planned combination to implemented status.

Control Room crosswalk¶

No dedicated equation editor exists. Use the applicable Geometry, Material, Physics, or Stage panel. Status: inspection-only for the scientific explanation. frontend support is not implemented applies to physical parameters without a matching control. See {doc}/frontend/capability-register; do not infer UI support from backend or Python availability.

Python/API crosswalk¶

The linked Python API page is authoritative for exact functions, arguments, units, and failure semantics. If this page is a foundation or category overview, runnable Python is ot applicable here and must be taken from the terminal API page.

Bibliography and source scope¶

Use the scientific bibliography and source-code index on the linked terminal page. This block adds no new equation or unverified implementation claim.

Source-code index¶