Including sample shape in micromagnetics with 3D periodic boundary conditions

Abstract: Periodic boundary conditions (PBCs) for computing magnetic fields in repeating magnetic structures, e.g. in micromagnetic simulations, are typically imposed using the quasi periodic macrogeometry approach, where many copies of the simulated domain are introduced. This can be computationally problematic, especially if the simulated domain is incommensurate with the desired sample shape. In this work, we present a formal proof that for sufficiently large magnetic samples, only the average magnetisation gives non-negligible shape effects. Using this insight, we develop a simple, computationally efficient modification of existing implementations which incorporates shape effects in PBC methods.

• The paper proves that, for sufficiently large samples with macroscopically uniform magnetisation, shape-dependent fields are determined by the average magnetisation and can be added through a low-cost demagnetisation correction.
• The proposed correction requires only two precomputed demagnetisation tensors and an O(n_cell) matrix-vector product per timestep, while reducing errors in 2D periodic simulations by nearly an order of magnitude and removing finite-size artifacts in 3D.
• The method captures aspect-ratio effects in 100 MHz composite-grain simulations, but it does not apply to macroscopically ordered structures such as Halbach arrays, whose fields can diverge logarithmically.

Overview

Durhuus, Insinga, and Bjørk address a well-known deficiency of periodic boundary conditions (PBCs) in three-dimensional micromagnetic simulations: the demagnetisation field depends on the shape of the entire sample through the demagnetisation tensor $\mathrm{N}$, which is volume-independent and therefore does not vanish as the simulated domain shrinks relative to the magnet. Standard PBC implementations that repeat the simulated domain to infinity implicitly impose an isotropic shape tensor, while finite macrogeometry approaches require many domain copies whose arrangement must match the sample aspect ratio. The paper provides a formal proof that, for sufficiently large samples with macroscopically uniform magnetisation, all shape-dependent contributions to the internal field are captured by the average magnetisation $\vb{M}_\text{avg}$ alone. Building on this result, the authors propose a computationally inexpensive "shape correction" field that can be added to any existing PBC scheme, and they validate it on a soft magnetic composite driven by a 100 MHz oscillating field.

Theoretical framework

The central decomposition splits the field from any region into two parts: a contribution from uniform magnetisation equal to the region's average magnetisation, and a residual $\Delta \vb{H}$ from local non-uniformity. Writing the field from each domain copy $\Gamma_i$ via the dipolar demagnetisation kernel and Taylor expanding about the copy center, the authors show that the leading term ($n=0$) integrates to zero because $\int_\Gamma \Delta \vb{M}\,\dd\vb{r} = 0$ by construction, so the non-uniform contribution scales as $r_i^{-4}$ rather than $r_i^{-3}$. A careful bounding argument — replacing sums over distant copies by integrals over spherical shells and proving absolute convergence of the resulting series via the Cauchy root test — establishes that

$\lim_{R_\Lambda \to \infty} \Delta \vb{H}_{\neg\Lambda} = 0,$

i.e., the field from locally non-uniform magnetisation converges to a shape-independent value even in 3D, where naive dipole sums are not absolutely convergent. Only the average-magnetisation term remains scale-invariant, consistent with the numerical observations of Matsuo and Yamazaki.

The practical consequence is that the total field inside the simulated domain is approximated as

$\vb{H}(\vb{r}) = \vb{H}_\Lambda(\vb{r}) - \mathrm{N}_{\overline{\Omega}}(\vb{r})\,\vb{M}_\text{avg},$

where $\vb{M}_\text{avg}$0 is the demagnetisation tensor of the hollow region between the macrogeometry and the full sample boundary; subtracting $\vb{M}_\text{avg}$1 avoids double counting the copies' average-magnetisation contribution already present in $\vb{M}_\text{avg}$2. For infinite 3D PBCs, translational and rotational symmetry give $\vb{M}_\text{avg}$3, so infinite PBCs are already correct only for spherical samples or isotropic bulk cubes; in all other cases the correction is necessary. Analytical demagnetisation tensors for prisms, cylinders, ellipsoids, and other geometries make evaluating the correction straightforward.

The computational overhead is modest: setup requires two additional per-cell demagnetisation tensors evaluated once, and each timestep adds an $\vb{M}_\text{avg}$4 matrix-vector product, versus the $\vb{M}_\text{avg}$5 cost of the macrogeometry sum itself. A further benefit is decoupling the macrogeometry shape from the true sample shape, granting flexibility when the desired sample geometry is incommensurate with the periodic tiling.

Proof assumptions and a counterexample

The proof relies on macroscopic uniformity of the magnetisation at scales beyond the simulated domain — an assumption implicit in all PBC methods but made explicit here. The authors construct a sharp counterexample: a spherical shell with magnetisation pointing up for polar angles outside $\vb{M}_\text{avg}$6 and down inside it has zero net moment yet produces a central field proportional to $\vb{M}_\text{avg}$7, diverging logarithmically regardless of how large the inner radius is. This logarithmic divergence mirrors Halbach arrays. The implication is stated plainly: for macroscopically ordered samples such as Halbach-type structures, standard demagnetisation tensors and PBC models are simply inapplicable, and no amount of shape correction rescues them. The authors note they cannot exclude other special orderings sharing the scale-invariance property, leaving this classification open.

A second structural limitation is that real magnets develop surface-localised deviations from bulk magnetisation near corners and edges, and surface material properties may differ; since PBCs presuppose uniformity above the domain scale, this inconsistency is inherent to the method rather than to the proposed correction.

Numerical validation

Magnetostatic convergence

Using MagTense, the authors simulate a cubic grain tiled with $\vb{M}_\text{avg}$8 cells under randomised magnetisation distributions with nonzero mean, comparing convergence of the demagnetisation field against a 15-layer reference for 2D and 3D PBCs, with and without the shape correction. Two results stand out:

• Mutual consistency: the mean discrepancy between the corrected and uncorrected references at $\vb{M}_\text{avg}$9 is only 0.01% for 3D and 2.6% for 2D PBCs.
• Dramatic improvement in 2D: without correction, 2D PBCs converge poorly because the slab-like macrogeometry misrepresents the sample aspect ratio at every finite $\Delta \vb{H}$0. With the correction, the error at $\Delta \vb{H}$1 drops by nearly an order of magnitude, and the gap widens with more layers. In 3D the improvement is small because the cubical macrogeometry already matches the sample shape; there the correction mainly removes a finite-size artifact by homogenising the average-magnetisation field.

This demonstrates that shape correction is essential for lower-dimensional PBCs and beneficial even when the macrogeometry nominally matches the sample.

High-frequency micromagnetics

The dynamical test case is a composite of cubic magnetic grains ($\Delta \vb{H}$2 nm, $\Delta \vb{H}$3 T, uniaxial anisotropy along $\Delta \vb{H}$4) in a sinusoidal field of amplitude $\Delta \vb{H}$5 T at 100 MHz along $\Delta \vb{H}$6, with sample aspect ratios $\Delta \vb{H}$7 and intergrain gaps from 0 to 1 μm. Key findings:

• At $\Delta \vb{H}$8 μm the magnetic volume fraction is only 0.086%, grains are effectively non-interacting, and no shape effect is observable.
• Despite zero expected coercivity from Stoner–Wohlfarth arguments (anisotropy axis perpendicular to the field), the observed coercive fields exceed the static bound $\Delta \vb{H}$9 mT by roughly a factor of ten. The authors attribute this to dynamical hysteresis: using the Steinmetz scaling $\Gamma_i$0 established for related systems ($\Gamma_i$1, $\Gamma_i$2), the coercive field must scale as $\Gamma_i$3, explaining large dynamic coercivity. These are, to the authors' knowledge, the first micromagnetic simulations of sample-shape effects in the high-frequency regime.
• For smaller gaps, shape effects emerge clearly: elongation along the field axis ($\Gamma_i$4) broadens and squarifies the loops by adding easy-axis shape anisotropy that raises the reversal cost, while $\Gamma_i$5 favours in-plane vortex states with gradual $\Gamma_i$6-magnetisation growth. Non-monotonous magnetisation changes and high-field loop openings at zero remanence appear for $\Gamma_i$7, attributed to high-frequency excitation of multi-domain states.
• For $\Gamma_i$8 with periodic exchange coupling, coercivity consistently increases due to stabilisation of the aligned state, though qualitatively the behaviour is unchanged.

Limitations and open questions

Several caveats bear directly on the applicability of the method. The formalism presumes macroscopically uniform magnetisation; macroscopically ordered configurations (e.g., Halbach arrays) produce logarithmically divergent fields that defeat both PBCs and the shape correction. The correction also assumes the sample is large enough that $\Gamma_i$9 converges before reaching the sample surface — for small entire samples, including $n=0$0 contributions from regions that do not exist introduces error. The treatment uses $n=0$1 throughout $n=0$2, valid only when the sample vastly exceeds the domain. Finally, the proposed extension to surface effects — solving the bulk problem first, then computing the surface demagnetisation field from $n=0$3 — is acknowledged to be non-self-consistent, since surface magnetisation feeds back into $n=0$4; its refinement and validation remain open.

Conclusion

The paper delivers a rigorous convergence proof showing that sample-shape effects in 3D micromagnetics reduce entirely to a scale-invariant demagnetisation field acting on the average magnetisation, provided magnetisation is macroscopically uniform. The resulting shape-correction field costs $n=0$5 per timestep, integrates trivially with existing PBC and macrogeometry schemes, restores shape physics to infinite-PBC simulations, and accelerates convergence of quasi-periodic macrogeometries — most strikingly for lower-dimensional PBCs, where errors fall by nearly an order of magnitude. Applied to a soft magnetic composite at 100 MHz, the method reveals that dynamic hysteresis dominates the response yet preserves the qualitative influence of aspect ratio on coercivity familiar from quasi-static magnetism.