Meshing and spatial discretization¶

Scope and purpose¶

Use this chapter before selecting an individual mesh generator. Fullmag separates four objects that are often conflated in simulation software:

• geometry: the continuous magnetic bodies and optional exterior universe;

• mesh intent: requested cell sizes, topology, selectors, size fields and build mode;

• mesh realization: immutable nodes, cells, facets, markers, quality and provenance;

• solver support: the element/cell families and boundary semantics accepted by the selected backend, device and physical interaction set.

A setting is production-safe only when all four agree. Authoring a prism request does not prove that prisms were generated; generating a valid CPU mesh does not prove that every GPU operator supports its cell families.

Scientific and numerical model¶

Cost models¶

For an FDM grid with \(N_xN_yN_z=N\) active lattice sites, local interactions scale approximately as \(O(N)\) and FFT demagnetization as \(O(N\log N)\), with padding and periodic-image choices contributing to memory. For FEM, assembly is approximately linear in the number of cells, while magnetostatic and linear-solver cost depend strongly on the airbox, polynomial order, conditioning and preconditioner.

The cheapest valid mesh is therefore not necessarily the one with the fewest entities. A geometry-conforming FEM mesh may reduce geometric error; a regular FDM grid may enable a much faster demagnetization operator. The choice must be tied to the observable and backend.

Selection guide¶

Parameters¶

Python API¶

Complete Python example

import fullmag as fm

nm = 1.0e-9


def build_fdm_study():
    study = fm.study("mesh_choice_fdm")
    study.engine("fdm")
    study.device("cpu", precision="double")
    study.mode("strict")
    study.universe(
        mode="manual",
        size=(160 * nm, 320 * nm, 10 * nm),
        center=(0.0, 0.0, 0.0),
        padding=(0.0, 0.0, 0.0),
    )
    study.cell(5 * nm, 5 * nm, 5 * nm)
    film = study.geometry(
        fm.Box(size=(100 * nm, 300 * nm, 10 * nm), name="fdm_film"),
        name="fdm_film",
    )
    film.Ms = 800.0e3
    film.Aex = 13.0e-12
    film.m = fm.texture.uniform(1.0, 0.0, 0.0)
    study.exchange()
    study.demag()
    study.stages.add_relax(stage_id="equilibrium", dt=5.0e-13, max_steps=10_000, tolA=1.0e-4)
    return study


def build_fem_study():
    study = fm.study("mesh_choice_fem")
    study.engine("fem")
    study.device("cpu", precision="double")
    study.mode("strict")
    study.universe(
        mode="manual",
        size=(500 * nm, 300 * nm, 160 * nm),
        center=(0.0, 0.0, 0.0),
        padding=(0.0, 0.0, 0.0),
    )
    study.universe.mesh(
        minimum_element_size=15 * nm,
        maximum_element_size=80 * nm,
        maximum_element_growth_rate=1.5,
        grading="geometric",
    )
    magnet = study.geometry(
        fm.Ellipsoid(110 * nm, 50 * nm, 20 * nm, name="fem_body"),
        name="fem_body",
    )
    magnet.mesh(
        mesh_strategy="free_tetrahedral",
        minimum_element_size=4 * nm,
        maximum_element_size=8 * nm,
        order=1,
        compute_quality=True,
    )
    magnet.Ms = 800.0e3
    magnet.Aex = 13.0e-12
    magnet.m = fm.texture.uniform(1.0, 0.0, 0.0)
    study.exchange()
    study.demag(realization="poisson_robin")
    study.stages.add_relax(stage_id="equilibrium", dt=5.0e-13, max_steps=10_000, tolT=1.0e-6)
    return study

Control Room workflow¶

1. Select Study and set the requested backend, device, precision and execution mode.

2. For FDM, author the structured grid in the Study Inspector. For FEM, select each object’s Mesh node and author an object-owned policy; select Universe / Airbox Mesh for the exterior-domain policy.

3. Apply policies before building. A policy edit makes older mesh resources stale by design.

4. Build the selected object mesh or the full shared-domain mesh through the canonical command.

5. Inspect requested vs. effective vs. realized values, quality scopes, markers and fallback evidence. Do not start the solver while the mesh resource is stale, degraded without approval, or incompatible with the active lane.

6. Repeat the simulation on a controlled refinement sequence and report convergence of the scientific observable.

Verification, quality and provenance¶

After every build, inspect the realized resource rather than assuming that the authored request was applied. The production check is:

• geometry and mesh revisions match the current model;

• requested and realized discretization/topology/order are recorded;

• node, element and boundary-facet counts are nonzero for every required region;

• region and boundary markers cover the complete topology;

• inverted and degenerate element counts are zero;

• interface diagnostics report no orphan, coincident, nonmanifold or unmatched facets;

• local size distributions are consistent with the intended edge/interface/core grading;

• any fallback or degradation has an explicit reason and an actual method;

• a mesh-refinement sequence demonstrates convergence of the scientific observable.

MeshQualityReport exposes signed inverse condition number (SICN), gamma/radius quality, volume statistics and optional per-element arrays. The source constants gamma_min=0.08 and SICN p05=0.1 are implementation gates for named report paths; they are not universal physical acceptance thresholds for every element family or study.

Mesh-convergence protocol¶

A production result should include at least three discretizations. Refine only the parameter under study while holding geometry, material parameters, solver tolerances, initial state and output sampling fixed. Let \(Q_h\) denote the observable for characteristic size \(h\). Report

with a documented scale for observables that can cross zero. For dynamics, compare resonance frequency, linewidth and mode profile; for relaxation, compare total energy and texture; for demag, compare field/energy and verify that moving the outer boundary does not change the result beyond the chosen tolerance.

Diagnostics and failure semantics¶

Treat the following as blocking unless a study-specific acceptance rule says otherwise:

• missing or stale mesh revision;

• unsupported cell family for the selected operator/device;

• requested/realized topology mismatch without an explicit fallback record;

• negative Jacobian, zero-volume cell or incomplete marker coverage;

• nonconforming magnetic/air interface in a conforming shared-domain solve;

• an FDM cell size that does not resolve the smallest active geometry or magnetic length scale;

• an airbox-convergence result that changes materially when the outer boundary is moved.

Where this is implemented¶

Implementation map reviewed against commit 5db00ccf0113b9756fec2d46feb36ade762b12c2 on 2026-08-24.

Related documentation¶

• FDM grids

• FEM shared domain

• Refinement and convergence

• Swept meshes

References¶

• C. Geuzaine and J.-F. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” International Journal for Numerical Methods in Engineering 79 (2009), 1309–1331, doi:10.1002/nme.2579.

• C. Abert, “Micromagnetics and spintronics: models and numerical methods,” European Physical Journal B 92, 120 (2019), doi:10.1140/epjb/e2019-90599-6.

• Gmsh reference manual, mesh algorithms, size fields, extrusion and physical groups: gmsh.info/doc/texinfo.

Documentation tree¶

Source-code index¶

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