GPU-accelerated Electromagnetic Open Differentiable Engine — FEM/DG
Status: driven + eigenmode FEM with multi-mode wave ports landed. End-to-end stack (mesh I/O → P1 + Nédélec kernels → autodiff-preserving sparse
[nnz]assembly → direct LU and Krylov COCG + ILU(0) solvers → PEC / Silver-Müller / matched UPML / Leontovich / lumped + multi-mode wave ports → S-parameter and NTFF extraction) is onmain, validated across five benchmarks: Mie sphere (eigenmode + driven scattering), patch antenna (S11 / bandwidth / NTFF / efficiency), spiral inductor (L/Q), and the SLCFET 3HP capstone. See Highlights below.
GEODE-FEM is a Burn-based Rust implementation of a high-order finite-element / discontinuous-Galerkin electromagnetic solver. It targets the same use cases as AWS Labs Palace — eigenmode analysis, frequency-domain driven simulation, time-domain — but built natively on a differentiable tensor IR with GPU acceleration as a first-class concern.
Palace is a state-of-the-art open-source FEM EM solver, but its C++/MFEM-based implementation predates the era of tensor IRs with differentiable, multi-backend GPU compilers. Expressing the same operators on top of Burn unlocks:
- Hardware portability without per-backend kernels: CUDA, ROCm, Metal, Vulkan, WebGPU
- Differentiable physics for inverse design and topology optimization
- Kernel fusion across FEM stencils via Burn's JIT compiler
- Rust end-to-end: solver, geometry, mesh, I/O — no Python boundary
GEODE-FEM is one of three complementary projects:
| Project | Role | Discretization | Status |
|---|---|---|---|
| crutcher/palace_whiteroom | Clean-room dissection of Palace into a layered specification (L1–L4) | FEM/DG (target) | Active analysis |
| rjwalters/geode-fem | Burn-based realization of the whiteroom L4 specification | FEM/DG | Driven + eigenmode; 5-benchmark suite; multi-mode wave ports |
| rjwalters/strata-fdtd | FDTD time-domain solver (acoustic today, EM in progress) | FDTD | Active |
Strata and GEODE-FEM are sister codebases that use different discretizations for overlapping physics. Mie resonances are the canonical cross-check benchmark — analytical Mie series ↔ strata FDTD ↔ GEODE-FEM eigenmode.
| Mie eigenmode | Lowest TM_1,1 mode (n=1.5 dielectric sphere, R/R_buffer=1/2, PEC outer + anisotropic UPML buffer): FEM Re(k) ≈ 1.229 vs analytic 1.30343, ~5.7% rel err / Q ≈ 27 on the bundled 774-node fixture. PR #60's anisotropic UPML (#54) broke the 16% scalar-PML reflection ceiling diagnosed in #52. |
| Mie driven scattering | Plane-wave scattered-field driven solve with matched (full Sacks) UPML; Q_ext via the volume optical theorem, Q_sca via Poynting flux, both vs the analytic Mie series. Below 5% rel err on the fine on-resonance fixture (#224); the coarse 774-node fixture is mesh-dominated at high ka. See benchmarks/mie_sphere/driven_results.toml. |
| Patch antenna | Probe-fed FR-4 patch (W/L/h = 38/29/1.6 mm, ε_r=4.4) with matched box-UPML and a Palace-style lumped probe port: f_res ≈ 2.27 GHz, S11 dip −15.2 dB, −10 dB BW 38.7 MHz (1.71%), η ≈ 29 % on the impedance-matched fixture (#237). NTFF via Love-equivalence yields directivity / gain (#229). vs Balanis cavity-model oracle (geode_core::patch_cavity). See benchmarks/patch_antenna/results_matched.toml. |
| Spiral inductor | 3.5-turn generic square spiral (54,428 edges) with Leontovich surface-impedance conductors: L extracted from Im(Z)/ω across a frequency sweep, within −4.9 % of Mohan analytic and −6.8 % of MoM PEEC at 1 GHz. See benchmarks/spiral_inductor/results.toml. |
| SLCFET 3HP capstone | 3-turn Au-on-SiC square spiral (76,964 edges) on the high-ε_r SiC substrate: f→0 quasi-static L₀ via Richardson extrapolation, −2.1 % vs MoM PEEC, +2.8 % vs Mohan — meeting the 5 % bar (#212). FEM correctly resolves substrate-C dispersion the analytic oracles omit. |
| Wave ports | Multi-mode wave-port BC with rank-N SMW augmentation and block S-matrix (A1+B1+C1+C2, #254/#255/#256/#257). 2D transverse modal eigensolver (#240, generalized to non-rectangular cross-sections in #265), bi-modal straight-section and height-step mode-matching cross-checks. |
| Iterative solver | Krylov COCG with Jacobi (#238) and ILU(0) (#267) preconditioners, wired through driven_frequency_sweep and solve_wave_port_sweep (#264). Sparse [nnz] Nédélec assembly (#218) lifted the prior 46k-edge dense-scatter cap. ILU(0) cuts iterations 2.4× on the σ-damped stress fixture vs Jacobi. |
| Math correctness | M_{ij} = ∫ N_i · N_j ε(x) dV is complex-symmetric (M^T = M), not Hermitian (M^H ≠ M) — the Mie inner product is bilinear, not sesquilinear. Caught during PR #55, validated by empirical check (Im(v^H M v) ≈ −58) and by-hand derivation. ILU(0) for COCG preserves the bilinear form (#267). |
| Validated chain | 200+ PRs merged. Scalar Helmholtz cube modes, batched P1 + Nédélec local kernels with autodiff through assembly, dense (faer::generalized_eigen) and sparse (shift-and-invert Lanczos, complex-symmetric variant for the Mie pencil) eigensolvers, ARPACK fallback driver (--features arpack), all four absorbing-BC families above, Palace 3D oracle slots for patch (#239) and spiral (#266). |
Each benchmark family ships a one-command tearsheet that overlays the FEM result on its analytic / empirical oracle. Regenerate any of them with:
python -m geode_viz.scripts.plot_benchmark <mie_sphere|spiral_inductor|patch_antenna|motor> --tearsheet(See tools/viz/ for the full plotting + VTK/ParaView export
pipeline — Epic #276.)
Mie sphere — scattering efficiencies Q_ext / Q_sca vs size parameter
ka, against the Bohren & Huffman analytic series, with a per-point
relative-error strip.
Spiral inductor — |S11| plus extracted L / Q / R vs frequency,
bracketed by the Mohan analytic band and the MoM PEEC range, with the
self-resonant frequency marked.
Patch antenna — the |S11| resonance dip, the input-impedance Smith
chart, and the E-/H-plane radiation-pattern cuts (dB) against the Balanis
cavity-model directivity oracle.
Slotless-PM motor — the locked-rotor torque-vs-angle capstone (Epic #448).
A radially-magnetized surface-PM rotor is driven by a θ-distributed stator
winding sheet J_z = J0 cos(pθ); the resulting PM-vs-stator interaction torque
T(θ_r) = −(2π L/μ₀)·C_M·G_S·cos(p θ_r) has an exact closed form, plotted here
against the FEM Arkkio (volume-averaged, preferred) and Maxwell line-integral
estimators over one electrical period. The Arkkio sweep tracks the analytic
oracle to 0.71 % L2 (37.6k-node mesh, ≤ 2 % target met), the error strip
staying under both the 2 % and 5 % guides — Arkkio's volume averaging cancels
the pointwise P1 product noise so the interaction torque clears the target even
though the raw mid-gap field sits at the documented ~2.4 % P1 floor. The
self-torques (pure-PM and pure-stator) are ≈ 0 by symmetry, so the graded number
is the genuine interaction torque, not a mesh-symmetry artifact.
Regenerate the benchmark fixture and field export with:
cargo run -p geode-core --release --example motor_torque
# → benchmarks/motor/results.toml (sweep) + artifacts/viz/motor/motor_field.vtuThe motor_field.vtu carries the nodal A_z scalar and the per-cell B glyph
vector for the θ_r = 0 cross-section. Render the ParaView machine-slice visual
(colour by A_z, add a Glyph filter on the cell vector B) with:
pvbatch tools/viz/geode_viz/scripts/pvbatch_render.py \
artifacts/viz/motor/motor_field.vtu --slice z=0 --field A_z --out motor_field_slice.pngThe driven solver can export its volumetric E(r) and the NTFF radiation
lobe as VTK .vtu files, rendered headlessly with ParaView's pvbatch:
# Near-field |E| on the patch resonance, then a z-slice render
cargo run -p patch_antenna --release -- --export-field --out-dir artifacts/viz
pvbatch tools/viz/geode_viz/scripts/pvbatch_render.py artifacts/viz/E_patch.vtu --out E_patch_slice.png
# 3D directivity lobe (open in ParaView, colour by D_dB)
cargo run -p patch_antenna --release -- pattern-3d --out-dir artifacts/vizNear-field |E| slice |
3D radiation lobe |
|---|---|
 |
 |
The slice resolves the TM₀₁ cavity mode — field maxima at the patch's two radiating edges — while the lobe shows the broadside main beam peaking at ≈ 5.6 dBi.
- Cargo workspace skeleton with Burn dependency
- Scalar Helmholtz on a tetrahedral mesh, vector Nédélec elements
- Dense + sparse complex-symmetric eigensolvers, PEC + Silver-Müller + scalar PML
- Mie sphere eigenmode benchmark vs analytic PEC-cavity catalog
- Anisotropic UPML (#54) — broke the 16 % scalar-PML reflection ceiling
- Vector-tracking k₀ (#48) — self-consistent Newton on the Silver-Müller pencil
- Matched (full Sacks) UPML lifted into Burn assembly + driven solve (#205, #223)
- Deterministic driven solve
A(ω)x = bwith volumetric current source (#194/#197) - Frequency sweep + extraction Z(ω) → L/R/Q/S11 (#209), N-port S-matrix (#219)
- Lumped ports + Leontovich BC for driven simulation (#206, #207)
- Driven Mie scattering benchmark Q_ext / Q_sca vs analytic series (#195/#200)
- Sparse
[nnz]Nédélec assembly for driven path — lifts the 46k-edge dense-scatter cap (#220) - Patch antenna benchmark S11 / bandwidth / NTFF / efficiency (#231–#237)
- Spiral inductor benchmark L/Q vs Mohan + MoM PEEC (#211/#225)
- SLCFET 3HP capstone Au-on-SiC spiral, quasi-static L₀ vs MoM within 5 % (#212/#230)
- Wave-port BC with rank-N SMW augmentation + block S-matrix (#234/#245)
- Multi-mode wave ports (A1 + B1 + C1 + C2) + analytic mode-matching (#254/#255/#256/#257)
- 2D transverse modal eigensolver for general cross-sections (#240/#265)
- Krylov COCG iterative solver + Jacobi (#238) and ILU(0) (#267) preconditioners
- Iterative path wired through sweep pipelines with
solver_modeknob (#264) - Palace 3D oracle integration for patch antenna (#239) and spiral inductor (#266)
- Visualization tooling — benchmark tearsheets + VTK
.vtufield export + headless ParaView render (Epic #276) - Whiteroom L4 mapping (#5) — operator-only tracker, ongoing
Requires Rust stable (1.96+, set in rust-toolchain.toml).
cargo build # builds workspace with default `wgpu` backend
cargo test # runs the GPU smoke test
cargo run --bin geode # prints backend / device / smoke resultgeode-core selects one Burn backend at compile time via mutually-exclusive
features:
# default — wgpu (Metal on macOS, Vulkan on Linux, DX12 on Windows)
cargo build
# CUDA (requires a CUDA toolkit and an NVIDIA GPU)
cargo build -p geode-core --no-default-features --features cuda
# Metal (Apple platforms only; local-only — no Apple CI runner)
cargo build -p geode-core --no-default-features --features metalEnabling more than one of wgpu / cuda / metal, or none of the four
backends, is a hard compile error — see the compile_error! guards in
crates/geode-core/src/lib.rs.
Note the macOS nuance: the default wgpu backend already runs on Metal at
runtime on macOS (wgpu selects the Metal graphics API there). The opt-in
metal feature is different — it pins the Metal graphics API and the MSL
(cubecl-msl) shader-compilation pipeline at compile time (burn::backend::Metal
= Wgpu<f32, i32, u8>), rather than going through wgpu's runtime adapter
selection. It is Apple-only and not exercised on CI (all runners are headless
Linux, which use the ndarray CPU backend); verify it locally on Apple hardware
with cargo run --bin geode --no-default-features --features metal.
The default build is pure Rust: backend GPU drivers (Metal, Vulkan, CUDA,
etc.) aside, no system Fortran/BLAS libraries are required. In particular,
sparse generalized eigensolves use a built-in shift-and-invert Lanczos
(SparseShiftInvertLanczos) that depends only on faer's sparse LU.
The opt-in arpack Cargo feature switches in an ARPACK-backed driver
(ArpackEigensolver) as a canonical reference alongside the in-tree
Lanczos. The Lanczos remains the default; ARPACK never becomes the
default by design (issue #24 non-goal). FFI bindings to dsaupd_c /
dseupd_c (the stable ARPACK ICB C wrappers, available since arpack-ng
3.7) are vendored inline in crates/geode-core/src/arpack.rs, so no
bindgen / clang / gfortran toolchain is required at build time —
the only build-time work is pkg-config-based discovery of the system
libarpack.
Install one of:
# macOS (Homebrew)
brew install arpack pkg-config
# Debian / Ubuntu
sudo apt-get install -y libarpack2-dev pkg-configThen build / test with the feature:
cargo build --features arpack -p geode-core
cargo test --features arpack -p geode-core --release \
--test sparse_eigensolver -- --ignoredIf pkg-config cannot find libarpack on your system, you can point
the build script at it manually:
ARPACK_LIB_DIR=/opt/homebrew/opt/arpack/lib \
cargo build --features arpack -p geode-coreSet ARPACK_STATIC=1 to request static linking (only useful if your
libarpack.a is in the search path; Homebrew ships the dylib only).
The Homebrew arpack formula does ship the ICB C headers under
$(brew --prefix arpack)/include/arpack/ and a working pkg-config file
— we use neither, since our FFI declarations are vendored. The header
story that motivated the original opt-in framing (a quirk in
arpack-ng-sys where its system feature can't resolve
<arpack/arpack.h> because Homebrew's arpack.pc sets includedir
one level too deep) is documented in crates/geode-core/src/arpack.rs.
crates/
geode-core/ # FEM kernels, assembly, eigensolvers, driven solve,
# ports / BCs, S-parameter + NTFF extraction,
# benchmark examples and integration tests
geode-util/ # pre-core staging layer above geode-core: shared
# math / convert / interop / fixture / viz helpers
# (Epic #414)
geode-cli/ # `geode` binary — prints device info and runs the smoke op
geode-validation/ # cross-backend reference tests (NumPy / JAX / Julia /
# ONNX / TF-Java) and analytic-oracle gates
Benchmarks under benchmarks/ (one subdirectory per fixture)
carry auto-generated results*.toml files paired with tests/ cases that
either compare within a documented tolerance band or skip-with-note when an
external oracle (e.g. Palace) is pending_operator_run.
geode-core depends on bunsen
(git-pinned, default-features = false, features = ["std"]) for
machine-checked tensor shape contracts: the define_shape_contract! /
unpack_shape_contract! / assert_shape_contract! family
(bunsen::contracts::*) — see crates/geode-core/src/assembly/p1.rs for the
template. New tensor-index / sequence code should prefer bunsen::ops
(float_arange, float_linspace, tensor ClampOp, repeat_interleave) over
hand-rolling the Burn-primitive equivalents. Note that ClampOp operates on
Tensor<B, D> only; scalar (non-Tensor) math is out of scope for this
convention.
Numerical baselines for the unit-cube Dirichlet Laplacian ground-mode
sweep are committed under
crates/geode-core/tests/fixtures/cube_convergence.toml. The values are
not analytic targets; they record what the current assembly +
faer eigensolver produces today. Their job is to catch unintended
regressions when assembly or the eigensolver change.
The diff-check test lives at
crates/geode-core/tests/cube_convergence_regression.rs. It is
#[ignore]d for the same reason as the other eigensolver tests:
faer 0.24's gevd::qz_real panics under debug-assertions. Run with:
# Run the regression diff-check (and all other ignored faer tests):
cargo test -p geode-core --release -- --ignored
# Run only the convergence regression:
cargo test -p geode-core --release \
--test cube_convergence_regression -- --ignoredIf an intentional change (e.g. mass-lumping, eigensolver swap) shifts
the per-level eigenvalues beyond the 1e-4 relative tolerance,
regenerate the fixture and commit it alongside the code change:
cargo run -p regen_cube_convergence_fixture --releaseCall out the regeneration in the PR description so reviewers know the baseline drift is intentional.
A criterion-based bench harness lives under
crates/geode-core/benches/. It establishes
a wall-clock baseline for the FEM pipeline so future performance
work has something to push against. The current numbers (Apple Silicon,
default wgpu backend) are committed to
benchmarks/perf/baseline.toml.
Reproduce the measurements:
# Runs all 5 benches; total wall-clock ≈ 25-30 min on M-series hardware,
# dominated by the Mie end-to-end (~70-90 s per sample × 10 samples).
cargo bench -p geode-coreCriterion writes per-bench HTML reports under target/criterion/
(gitignored). Extract a clean TOML summary (medians + median-absolute-
deviation as an IQR proxy) with:
cargo run -p extract_baselineThis walks target/criterion/<bench>/<input>/new/estimates.json and
overwrites benchmarks/perf/baseline.toml. The extractor is not
wired into cargo bench itself — re-running the analysis is then a
side-effect-free second step.
Per-stage cost (n=10 cube; May 2026 baseline, pre-sparse-[nnz]):
| stage | median |
|---|---|
assemble_global_p1 |
45 ms |
assemble_global_nedelec (real) |
289 ms |
assemble_global_nedelec (cmplx) |
407 ms |
FaerDenseEigensolver |
5.95 s |
SparseShiftInvertLanczos |
52 ms |
Dense generalized_eigen dwarfs every other stage by ~100×. The
pure-Rust sparse shift-and-invert Lanczos (faer sparse LU) brings the
eigensolve down to roughly the same order as the Burn-side assembly.
Mie sphere end-to-end (774-node refined fixture, complex pencil):
| solver path | median |
|---|---|
FaerComplexEigensolver (dense) |
126.1 s |
SparseComplexShiftInvertLanczos (sparse) |
4.07 s |
31× speedup at this scale; 107× on the original 313-node fixture.
The sparse path is now the default in examples/mie_sphere.rs; pass
--dense for the correctness-oracle cross-check.
Scaling beyond the dense-scatter cap. The numbers above are from
the original dense Nédélec scatter path. Sparse [nnz] pattern-slot
assembly (#220) lifted the ~46 k-edge ceiling that pipeline hit, so
the driven path (driven_solve, driven_frequency_sweep,
solve_wave_port_sweep) now exercises fixtures into the 10⁵-edge
range — the spiral inductor (54 k edges) and SLCFET 3HP (77 k edges)
benchmarks run on it directly. The Krylov COCG iterative solver
(#243) and ILU(0) preconditioner (#267) provide a memory-frugal
alternative for fixtures the direct LU can't factor; both are
wired through the sweep pipelines under a solver_mode knob (#264).
The per-stage cost table above will be regenerated as part of the
next perf-baseline refresh; for now it stands as a known-stale but
documented reference point.
The project's stated north-star validation problem: FEM eigenmodes of a
dielectric sphere (refractive index n = 1.5, radius R = 1) inside a
vacuum buffer (r ≤ R_buffer = 2) terminated by an anisotropic UPML
(issue #54, default since issue #61), compared against analytic resonance
roots. The legacy scalar-isotropic PML is retained as --scalar-pml for
cross-check.
Run the benchmark:
cargo run -p mie_sphere --release # anisotropic UPML, sparse (defaults)
cargo run -p mie_sphere --release -- --dense # anisotropic UPML, dense oracle
cargo run -p mie_sphere --release -- --scalar-pml # legacy 16% baseline cross-checkThis prints a comparison table and writes
benchmarks/mie_sphere/results.toml
with the lowest 8 FEM modes paired against the extended analytic
catalog. The benchmark uses the PEC-cavity dielectric resonator
as the analytic ground truth (a closed cavity with PEC at r = R_buffer,
which is the limit the FEM hits as the PML absorption strength σ₀ → 0);
the open-space Mie WGM positions — which require Hankel functions and
complex Newton iteration — are tracked under #33. The driven
scattering (Q_ext, Q_sca vs. ka) cross-check is the companion
benchmark examples/mie_driven_scattering.rs (issue #195), which
writes
benchmarks/mie_sphere/driven_results.toml.
Mesh: bundled 774-node / 3335-tet fixture (tests/fixtures/sphere.msh,
regenerated from mesh_scripts/sphere.geo via Gmsh CLI). Layered into
sphere_interior (r ≤ 1) + vacuum_gap (1 < r ≤ 1.5) + pml_shell
(1.5 < r ≤ 2) + boundary triangles.
Catalog: roots for angular orders l ∈ [1, 4], both TE and TM
polarisations, lowest 5 radial overtones each (~40 entries). Each
root carries its (l, n, polarisation, multiplicity = 2l+1) label.
Mode classification: walks the catalog in ascending k and for
each analytic root claims the next 2l + 1 consecutive FEM modes
(sorted by Re(k)), producing an unambiguous (l, n, pol, m_idx)
label per mode. On the bundled fixture (anisotropic UPML default)
this identifies the lowest 3 FEM modes as the TM_1,1 triplet
(Q ≈ 27) and the next 3 as TE_1,1 (rel err ≲ 2.5%).
Current numbers (bundled fixture, anisotropic UPML default, σ₀ = 5.0, k₀_ref = 2.0):
| mode | analytic kR | FEM Re(kR) | rel err Re(k) | Q |
|---|---|---|---|---|
| TM_1,1 | 1.30343 | ≈ 1.229 | ≈ 5.7% | ≈ 27 |
| TE_1,1 | 1.88943 | ≈ 1.872 – 1.934 | ≈ 0.7 – 2.3% | ≈ 9 – 50 |
For comparison, the legacy --scalar-pml path produces TM_1,1 at
~16.2% rel err / Q ≈ 5.8 — the h-independent reflection floor
diagnosed in issue #52 and broken by issue #54's anisotropic UPML.
Why anisotropic helps (issue #52 → #54). Under the scalar-isotropic
PML the TM_1,1 / TE_1,1 / TM_2,1 modes ALL sat at ~16% rel err
independent of mesh refinement — the signature of an h-independent
reflection floor at the inner PML interface, not a discretization
error. A diagonal anisotropic permittivity tensor in the global
Cartesian basis, ε_α = (1/s_r) r̂_α² + s_t (1 - r̂_α²) per centroid
radial unit vector r̂, absorbs along the propagation direction in
a direction-aware way and removes the reflection floor. Available via
[assemble_global_nedelec_with_anisotropic_epsilon] and
[build_anisotropic_pml_tensor_diag]; default in examples/mie_sphere.rs
since issue #61.
On the "full rotation" follow-up. For the current PML profile
s_r = s_t = 1 - jσ/ω the off-diagonal terms of the rotated tensor
R · diag(1/s_r, s_t, s_t) · R^T are identically zero, so the
diagonal-only kernel is mathematically exact (not an approximation)
for this profile. The full off-diagonal kernel only matters when the
radial and tangential profiles diverge (e.g., CFS-PML or split-field
formulations), and is tracked as a future ticket against those
profiles, not against the present implementation.
Vector-tracked self-consistent k₀ (issue #48): the Silver-Müller
absorbing BC matches its impedance to a single guess k₀; the
resulting Q is dominated by impedance mismatch when k₀ is far from
Re(k_target). PR #47 added a damped Newton iteration with a frozen
integer index, which improved Q only marginally (≈ 0.54) because the
177-mode Whitney spurious cluster re-shuffles as k₀ drifts and the
integer-index target drifts off the physical mode. The vector-tracked
variant (self_consistent_k_vector_tracked in
silvermuller_self_consistent.rs) instead selects, at every Newton
iteration, the mode whose eigenvector has maximum bilinear-M
overlap with the prior iteration's target — metric-consistent with the
complex-symmetric mass matrix. Mode death is surfaced as a dedicated
SelfConsistentResult::ModeLost variant when the maximum overlap
falls below 0.5 (signalling the seed is far from any physical
resonance). This is the unblocker for tightening Q-factor agreement
on the Silver-Müller path; full convergence and Q comparison runs
live under tests/silvermuller_self_consistent_vector_tracking.rs
(#[ignore]'d, faer-dense-release-only).
The same physical problem is computed in the time domain by the sister
project rjwalters/strata-fdtd
via FDTD; eigenfrequency-level cross-validation across the two
discretizations is the goal of this benchmark family.
The acceptance test (crates/geode-core/tests/mie_sphere.rs) asserts
(a) the lowest FEM mode's Re(k) agrees with the analytic TM_1,1
root to within 8% at the bundled fixture's resolution under the
anisotropic-UPML default (tightened from 25% per issue #61 after
#54 broke the scalar 16% ceiling), and (b) the lowest TM_1,1
triplet's median Q is above 1.5 — a regression catch for PML
mis-configuration (σ₀ drift, mask break, vacuum-gap removal). The
present median Q on the anisotropic path is ≈ 27.
MIT