FEM → FDM state transfer¶

Scope and purpose¶

This page specifies the source-backed continuation from a FEM nodal state to target FDM cell centres, including element location, interpolation, normalization and failure reporting.

Scientific and numerical model¶

The target grid is sampled geometrically from the source FEM topology; this operation does not claim equivalence of the FEM and FDM operators or trajectories.

Physical and numerical problem¶

When a FEM continuation state is used to start an FDM stage, nodal FEM magnetization is evaluated at target FDM cell centres. This is a geometric field transfer, not a solver conversion: the target grid, cell-centre convention, outside-domain policy and unit-vector renormalization are part of the resolved operation.

Governing equations¶

For a target cell centre \(\mathbf x_i\), locate the containing source element \(T_i\) and evaluate the P1 field using barycentric weights \(\lambda_a\):

For magnetization, interpolation can change the norm, therefore the continuation path applies

Symbols and SI units¶

Assumptions and validity¶

• Source topology and nodal field ordering must match. Cell centres outside every source element are reported as outside/fallback; they are not silently extrapolated as valid FEM values.

• P1 interpolation is exact for affine fields inside a simplex, not for arbitrary higher-order or discontinuous fields.

• Renormalization preserves direction but changes the interpolated amplitude; record this whenever magnetization is transferred.

Python API¶

There is no separate user-facing transfer_fem_to_fdm() constructor. The operation is automatically performed by the runtime when a FEM continuation artifact feeds an FDM stage. The author still uses the normal stage-first study contract:

# %% Stage-first target FDM continuation
import fullmag as fm

nm = 1.0e-9
study = fm.study("fdm_continuation_target")
study.engine("fdm")
study.device("cpu", precision="double")
study.mode("strict")
study.objects.mesh.defaults(cell_size=(2 * nm, 2 * nm, 5 * nm))
film = study.geometry(fm.Box(size=(100 * nm, 20 * nm, 5 * nm), name="film"), name="film")
film.Ms = 8.0e5
film.Aex = 1.3e-11
film.m = fm.init.UniformMagnetization((1.0, 0.0, 0.0))
study.stages.add_relax(stage_id="continue", algorithm="nonlinear_cg", tolT=1.0e-6, max_steps=100)

Parameters¶

The resolved parameters are the source FEM topology, target FDM grid, outside-domain policy and normalization policy recorded by the continuation operation; there is no separate public transfer constructor.

ProblemIR and provenance¶

The transfer is represented in runtime continuation metadata, not as a new energy term. Record source mesh digest, target grid origin/cell size/dimensions, interpolated/outside counts, normalization policy, fallback count and source/target backend identity.

Diagnostics and failure semantics¶

Diagnostics must distinguish located and outside target points, interpolation failures, normalization changes, field-length mismatches and source/target artifact identities. Missing topology or invalid grid metadata is a validation error rather than an implicit fallback.

Round-trip and failure semantics¶

Requested intent is the requested continuation backend sequence; resolved execution is the actual source/target transfer result. Validation errors include missing topology, incompatible field length, invalid target grid and transfer failure. Unsupported combinations are reported with outside/fallback counts; no silent extrapolation is allowed.

Discrete realization by lane¶

Implementation mapping¶

Validation¶

Use affine-field exactness, uniform-field preservation, outside-domain accounting, norm statistics, and continuation energy/torque checks on the first target stage. Report all transfer counters.

Limitations¶

The transfer is interpolation, not conservative projection of arbitrary FEM quantities. It does not prove that FEM and FDM discretizations have identical energy or demagnetizing fields.

Scientific bibliography¶

• Standard P1 finite-element interpolation and barycentric coordinates reference.

• Canonical continuation implementation is listed below.

Control Room workflow¶

Author the target FDM grid and continuation stage in Control Room, then inspect the resolved grid and transfer counters before execution. Only controls surfaced by the current stage draft are authorable.

Control Room crosswalk¶

Use Model Explorer -> Stages -> Add stage -> <stage kind> for stage-level controls when the terminal page identifies a matching field. The current editor is partial: only fields surfaced by the stage draft are authorable. Numerical parameters without a matching control are not implemented in the frontend. Do not infer frontend support from Python or backend availability. See {doc}/frontend/capability-register for the current register and exact source owner.

Where this is implemented¶

The source-code index below records the stable routing and interpolation declarations used by this page and its sidecar map.

Source-code index¶