Accelerated electrical, thermal, EMC, and coupled PCB analysis

Accelerated PEEC and matrix-free FEM solvers for electrical and thermal PCB analysis, radiated-emission evaluation for EMC, and PDN optimization.

The repository holds two solver families under the electrical package, one under the thermal package, one evaluation front end under emc, and the coupled scenarios that chain them under multiphysics. electrical.dice_peec provides exact incremental delta-scoring for local PCB reroutes, a 2.5D multilayer interaction operator, a thin-sheet PEEC field solver, and an adaptive runtime controller for topology-optimization loops. electrical.matrix_free_mpir_fem provides a matrix-free Q1 finite-element solver for DC conduction and 2D frequency-domain Maxwell fields, driven by mixed-precision iterative refinement (MPIR) on NumPy or CuPy. thermal.matrix_free_mpir_fem solves steady heat conduction through the whole board stack with the same MPIR solver and runtimes, and takes the Joule loss of an electrical solve as its heat load. emc.tiled_dipole_superposition reads the current distribution of either electrical solve as Hertzian dipoles and evaluates near-field scans, far-field patterns, radiated power, and margins to CISPR 32 and FCC Part 15 limits. multiphysics.staggered_coupling runs the scenarios that couple them: the electro-thermal fixed point with temperature-dependent copper, and the emission of a cold or thermally converged current distribution.

Features

• Exact sparse delta scoring — O(s) + O(s²) incremental evaluation instead of O(N log N) full FFT per candidate
• 2.5D multilayer operator — planar FFT with interlayer kernel matrix and frequency-dependent via model
• CUDA acceleration — CuPy/cuFFT backend with RawKernel batched scoring
• Adaptive controller — memory-bounded, error-driven fidelity cascade with automatic OOM recovery
• Low-memory mode — near/far field splitting for 4–8 GB VRAM devices
• Matrix-free MPIR-FEM — Q1 PCB conduction FEM with an FP32 inner PCG and FP64 outer reliable updates
• Frequency-domain Maxwell — 2D scalar-polarisation wave, dielectric-loss, eddy-current, and skin-effect solve with complex64 inner GMRES and complex128 reliable updates
• CUDA matrix-free FEM — fused, node-owned Q1 gather kernel plus batched complex64 GMRES; no assembled matrix and no global atomics
• Accelerator boundary — NumPy/CuPy runtimes with a narrow low-precision interface that remains suitable for the later Tenstorrent port
• Thermal MPIR-FEM — matrix-free trilinear hexahedral heat conduction through the copper/laminate stack with anisotropic laminate conductivity, convective faces, fixed-temperature nodes, and a closed heat budget
• Electrothermal coupling — per-element and per-via Joule loss from the DC solve mapped onto the thermal stack without interpolation
• Radiated emissions — exact Hertzian-dipole superposition of the solved currents for near-field scans and far-field patterns, radiated power, dipole-moment proxies, PEC ground-plane images, and margins to CISPR 32 / FCC Part 15 limit lines, tiled on NumPy or CuPy
• Coupled scenarios — staggered electro-thermal iteration with σ(T) and via R(T), warm starts and Aitken relaxation; electro-emission and electro-thermal-emission chains; one run_scenario entry point

Installation

From PyPI

pip install pcb-analysis
# with CuPy and PyPEEC
pip install 'pcb-analysis[cuda]'

Each release is uploaded to PyPI by the Release workflow with Trusted Publishing, and the same wheel, source distribution, and SHA-256 checksums are attached to the GitHub release. To install a specific release without PyPI, point pip at the release asset or at the tag:

pip install https://github.com/AFLOY/pcb-analysis/releases/download/v0.6.0/pcb_analysis-0.6.0-py3-none-any.whl
pip install 'git+https://github.com/AFLOY/pcb-analysis.git@v0.6.0'

The release tag has to match the version in pyproject.toml.

From a checkout, CPU only

pip install -e .

With CUDA support

pip install -e '.[cuda]'

Note: Requires an NVIDIA driver and CUDA 13.x. Verify with nvidia-smi. The package is installed as electrical; the former top-level peec_fastopt package now lives at electrical.dice_peec, and the thermal and EMC front ends and the coupled scenarios are the separate top-level packages thermal, emc, and multiphysics. Re-run the editable install after pulling these changes so the new packages are importable and the old path is not left on sys.path.

Quick start

from electrical.dice_peec import (
    Stackup,
    FFTInteraction25D,
    MultilayerDeltaScorer,
    CandidateEdit,
    SegmentOp,
    ViaOp,
    compile_candidate,
)
import numpy as np

stack = Stackup.dual_sided(board_thickness_mm=1.6)
shape = (64, 64)
base = np.zeros((stack.n_layers, *shape))
base[0, 10:20, 10:30] = 1.0

op = FFTInteraction25D(shape, stack, cell_size_m=0.2e-3)
scorer = MultilayerDeltaScorer(op, base, frequency_hz=3e5)

edit = CandidateEdit(
    segments=[SegmentOp("add", "B.Cu", ((12, 12), (12, 13), (12, 14)))],
    vias=[ViaOp("add", 12, 14, "F.Cu", "B.Cu", important=True)],
)
compiled = compile_candidate(edit, stack)
score = scorer.energy(compiled.occupancy_delta, vias=compiled.vias)

Matrix-free PCB conduction

import numpy as np
from electrical.matrix_free_mpir_fem import (
    CurrentTerminal,
    LayeredPCBMesh,
    PCBConductionProblem,
    solve_pcb_dc,
)

mesh = LayeredPCBMesh(
    element_active=np.ones((1, 1, 32), dtype=bool),
    layer_thickness_m=(35e-6,),
    pitch_x_m=0.2e-3,
    pitch_y_m=0.2e-3,
)
problem = PCBConductionProblem(
    mesh=mesh,
    terminals=(
        CurrentTerminal(((0, 0, 0), (0, 1, 0)), 1.0, "source"),
        CurrentTerminal(((0, 0, 32), (0, 1, 32)), -1.0, "sink"),
    ),
    reference_node=(0, 0, 32),
)
solution = solve_pcb_dc(problem)
assert solution.solve.converged

Frequency-domain Maxwell

The frequency-domain front end solves the E_z Maxwell reduction for z-invariant geometries and reports electric field, magnetic field, eddy-current density, conductor loss, and dielectric loss. Dirichlet nodes are given by a boolean mask over the node grid plus the field value held at each masked node.

import numpy as np
from electrical.matrix_free_mpir_fem import (
    ScalarMaxwellMesh2D,
    ScalarMaxwellProblem,
    solve_scalar_maxwell,
)

mesh = ScalarMaxwellMesh2D(
    (8, 64),                # elements: rows x columns
    pitch_x_m=0.1e-3,
    pitch_y_m=0.1e-3,
    relative_permittivity=4.0,
    dielectric_loss_tangent=0.01,
)
mask = np.zeros(mesh.node_shape, dtype=bool)
mask[:, 0] = mask[:, -1] = True          # drive the left edge, ground the right
value = np.zeros(mesh.node_shape, dtype=np.complex128)
value[:, 0] = 1.0
problem = ScalarMaxwellProblem(mesh, 1e9, mask, value)

solution = solve_scalar_maxwell(problem)          # NumPy complex64 inner solve
assert solution.solve.converged
loss_w_per_m = solution.dielectric_loss_w_per_m

CUDA full-wave solve

solution = solve_scalar_maxwell(
    problem,
    backend="cuda",  # or "auto" to fall back to NumPy when no GPU is visible
    device_id=0,
)
assert solution.solve.low_runtime == "cupy-complex64"

The complex128 reliable residual stays on the CPU. The correction RHS crosses to CUDA once per outer iteration, while the complex64 Krylov basis, preconditioner, material coefficients, and repeated Q1 actions remain on the GPU. On the repository's GTX 1650 audit, the fused operator is 8.8–40.4× faster than the NumPy complex64 action for 4,225–66,049 unknowns. The complete solve is slower at 4,369 unknowns (0.85×) but 1.99× faster at 16,705 unknowns, so CUDA should not be selected solely for tiny meshes.

Steady thermal conduction

The thermal front end meshes the whole stack as slabs of hexahedral Q1 elements. Copper, laminate, and vias are element conductivities; laminates may have a different through-plane value. Heat enters per element or per node and leaves through convective faces and fixed-temperature nodes. The solution reports nodal temperature, element heat flux, and a heat budget whose error is the solver residual.

import numpy as np
from thermal.matrix_free_mpir_fem import (
    ConvectionBoundary,
    HeatSource,
    LayeredThermalMesh,
    ThermalConductionProblem,
    solve_thermal_conduction,
)

# 35 µm copper / 1.5 mm FR-4 / 35 µm copper, 50 mm x 50 mm on a 0.5 mm grid.
mesh = LayeredThermalMesh(
    slab_thickness_m=(35e-6, 1.5e-3, 35e-6),
    pitch_x_m=0.5e-3,
    pitch_y_m=0.5e-3,
    conductivity_w_per_m_k=(385.0, 0.8, 385.0),
    through_plane_conductivity_w_per_m_k=(385.0, 0.3, 385.0),
    element_shape=(100, 100),
)
ambient = 298.15
problem = ThermalConductionProblem(
    mesh,
    convection=(
        ConvectionBoundary("top", 10.0, ambient),
        ConvectionBoundary("bottom", 10.0, ambient),
    ),
    heat_sources=(HeatSource(((3, 50, 50), (3, 50, 51)), 0.5, "regulator"),),
)
solution = solve_thermal_conduction(problem, initial_temperature_k=ambient)
assert solution.solve.converged
hottest_k = solution.max_temperature_k
assert abs(solution.heat_balance_error_w) < 1e-9

The default preconditioner adds a patch-constant coarse correction to Jacobi scaling. A cooled copper plate is stiff in-plane and weakly coupled to the air, so plain Jacobi PCG needs hundreds of inner iterations per decade on such a stack; the two-level variant needs tens. Pass preconditioner="jacobi" to compare, and coarse_block_nodes to set the patch width.

Electrothermal coupling

element_joule_heat_w places the per-element copper loss of a PCBConductionSolution onto the thermal slabs that hold each electrical layer, and via_joule_heat_sources turns each via's loss into nodal heat at its endpoints. Both meshes must share the in-plane element grid.

from thermal.matrix_free_mpir_fem import element_joule_heat_w, via_joule_heat_sources

electrical = solve_pcb_dc(pcb_problem)          # copper on layers 0 and 1
layer_slabs = (0, 2)                            # bottom and top copper slabs
thermal_problem = ThermalConductionProblem(
    mesh,
    convection=(ConvectionBoundary("top", 10.0, ambient),),
    element_heat_w=element_joule_heat_w(electrical, mesh, layer_slabs),
    heat_sources=via_joule_heat_sources(pcb_problem, electrical, mesh, layer_slabs),
)
solution = solve_thermal_conduction(thermal_problem, backend="auto")

The thermal operator runs its FP32 inner PCG on CuPy with backend="cuda"; the FP64 reliable update stays on the host exactly as for the electrical solves.

Radiated emissions

Every current element of a solved board is a Hertzian dipole. The EMC front end sums their exact fields for a near-field scan and their far-zone terms for the pattern at a test-site distance, then compares the maximum with a limit line.

import numpy as np
from emc.tiled_dipole_superposition import (
    CISPR32_CLASS_B,
    dipole_moments,
    dipoles_from_pcb_dc,
    emission_margin,
    evaluate_fields,
    far_field_pattern,
    scan_plane,
)

electrical = solve_pcb_dc(pcb_problem)            # two copper layers
dipoles = dipoles_from_pcb_dc(
    pcb_problem, electrical, layer_height_m=(0.0, 1.6e-3), close_terminals=True
)
frequency_hz = 100e6                              # DC pattern used as a phasor

probe = scan_plane(np.linspace(0, 0.05, 50), np.linspace(0, 0.02, 20), z_m=6.6e-3)
near = evaluate_fields(dipoles, probe, frequency_hz, backend="auto")
h_max_a_per_m = near.magnetic_magnitude_a_per_m.max()

pattern = far_field_pattern(dipoles, frequency_hz, distance_m=10.0)
margin = emission_margin(
    pattern.max_polarised_field_v_per_m, frequency_hz, CISPR32_CLASS_B, distance_m=10.0
)
print(pattern.radiated_power_w, margin.predicted_dbuv_per_m, margin.margin_db)

moments = dipole_moments(dipoles, frequency_hz)   # |P| ≈ 0 once the loop closes

dipole_moments reports the net electric moment: a non-zero value for a board whose terminal currents balance means the return path through the component is not in the model, and close_terminals=True adds it as a straight element. The magnetic near field and the far field are the robust outputs; the electric near field of a current-only description is sensitive to how continuously the element chain closes. dipoles_from_sheet_peec does the same for a frequency-resolved sheet-PEEC solve, without the quasi-static assumption.

Coupled scenarios

Scenario dataclasses name what is coupled to what; run_scenario dispatches on the type. The electro-thermal chain iterates the two conduction solves until the copper conductivity σ(T) = σ_ref / (1 + α (T − T_ref)) is self-consistent with the temperature it produces, warm-starting both solves and relaxing the temperature update with Aitken's Δ² estimate.

from multiphysics.staggered_coupling import (
    ElectroThermalEmissionScenario,
    ElectroThermalScenario,
    EmissionScenario,
    run_scenario,
)

coupled_scenario = ElectroThermalScenario(
    pcb_problem,                    # PCBConductionProblem, conductivity at 293.15 K
    thermal_mesh,                   # LayeredThermalMesh on the same (rows, cols)
    layer_slabs=(0, 2),             # thermal slab holding each copper layer
    convection=(ConvectionBoundary("top", 10.0, 298.15), ConvectionBoundary("bottom", 10.0, 298.15)),
)
coupled = run_scenario(coupled_scenario)
print(coupled.converged, coupled.iterations, coupled.loss_increase_ratio)

chained = run_scenario(
    ElectroThermalEmissionScenario(
        coupled_scenario,
        layer_height_m=(0.0, 1.6e-3),
        emission=EmissionScenario((30e6, 100e6, 300e6)),   # CISPR 32 Class B at 10 m
    )
)
print(chained.emission.margin_db, chained.heating_shift_db)

ElectricalScenario, ThermalScenario, ElectroEmissionScenario, and SheetPeecEmissionScenario (one sheet-PEEC solve per frequency) complete the set; run_scenarios runs a list. See MULTIPHYSICS_SCENARIOS.md.

Project structure

src/
  electrical/                         Analysis target
    dice_peec/                         PEEC + DICE/2.5D FFT acceleration
    matrix_free_mpir_fem/              FEM + matrix-free/MPIR acceleration
  thermal/                            Analysis target
    matrix_free_mpir_fem/              Heat-conduction FEM + matrix-free/MPIR acceleration
  emc/                                Analysis target
    tiled_dipole_superposition/        Radiated emission by dipole superposition, tiled NumPy/CuPy
  multiphysics/                       Coupled analysis
    staggered_coupling/                Partitioned fixed-point coupling with warm starts and Aitken relaxation
tests/                                 Test suite (pytest)
docs/                                  Design documents, validation reports, raw results
examples/                              Demo scripts and benchmarks
experiments/                           Repeatable accuracy and timing audits
requirements.txt                       Editable install with CUDA/test/build tooling

Future physics should follow the same rule: the first level names the analysis target and the second names the method plus its acceleration strategy.

Electrical solvers

cuda_pypeec runs PyPEEC's own three-dimensional voxel solve on CUDA. PyPEEC owns the physics; this owns the device policy. pypeec_memory predicts what a model costs before it is attempted, which matters once a model spans a board's height instead of one copper layer: on a two-sided 34.7mm board the prepared operators go from 4.5 MiB to 541 MiB while the conductor merely doubles.

sheet_peec solves the same physics on a mesh built for what a PCB is -- a few thin sheets at heights the stackup states. Its inductance operator is a two-dimensional transform per layer pair rather than a three-dimensional one over the board's height, which brings the same board's operators to 5.6 MiB. At zero frequency it reproduces an independent resistor network to machine precision; above it, the transform path matches a dense assembly of the same operator. See docs/SHEET_PEEC.md.

multilayer_peec is neither. It is a scalar interaction proxy for ranking many candidate shapes cheaply, and it solves for no current or potential.

skin_filaments extends sheet_peec to conductors thick against the skin depth by cutting them into graded filaments joined through the thickness, so the solve itself decides how the current divides between the faces and the interior.

matrix_free_mpir_fem solves real SPD DC conduction on layered Q1 PCB meshes and complex 2D scalar-polarised frequency-domain Maxwell fields. Its global matrix is never assembled. Low-precision inner corrections contain most operator applications, while high-precision host residuals determine final convergence. Scope and the Blackhole port boundary are documented in docs/MATRIX_FREE_MPIR_FEM.md.

Experiments and audits

The scripts in experiments/ regenerate the numbers quoted in docs/. They run from the repository root because the FP32 audit is also imported by the test suite. Results default to the ignored benchmark-results/ directory; pass --output (or --json for the FP32 audit) to write elsewhere.

# What single precision costs the sheet-PEEC solver (host stage, then CUDA)
python experiments/fp32_accuracy.py --cells 128 --json benchmark-results/fp32.json

# Maxwell accuracy against closed-form 1D solutions, plus CPU timing
python experiments/maxwell_small_benchmark.py

# Fused CUDA operator throughput and complete-solve timing (needs a GPU)
python experiments/maxwell_cuda_benchmark.py --operator-sides 64 128 256

# Sheet PEEC versus matrix-free FEM on matched DC strips and a skin-effect slab.
# Both methods solve both scenarios; the PEEC skin bar takes ~90 s at 64 cells.
python experiments/peec_fem_comparison.py --no-cuda --peec-skin-lengths 16 32

The skin-effect scenario is the same 0.5 mm copper slab at 1 MHz for both methods. FEM solves the field through the slab's thickness. Sheet PEEC has no infinite slab, so it solves a bar of graded filaments and reads the AC/DC resistance ratio from the filament current division at the middle of the bar, where it is slab-like. The closed-form slab impedance is the shared reference.

Development

# CPU-only, as CI runs it
pip install -e '.[test]'

# Or: the ignored local environment with CUDA, test, and build tooling
python -m venv .venv
source .venv/bin/activate
python -m pip install -r requirements.txt

# Run tests from the repository root
python -m pytest tests/ -v

CI runs the suite on Python 3.11–3.13, then builds a wheel and source distribution with the oldest supported setuptools and imports both subpackages from a clean environment, so a module missing from the distribution fails the build rather than a user's install.

Documentation

• Architecture & Design
• CUDA Backend Handoff
• Matrix-free MPIR-FEM
• Thermal MPIR-FEM
• EMC radiated emissions
• Coupled analysis scenarios
• Sheet PEEC and Sheet PEEC CUDA results
• Requirements
• Sheet PEEC / matrix-free FEM comparison (raw data)
• Maxwell accuracy and CPU timing report (raw data)
• Maxwell CUDA benchmark data and CUDA scaling data
• Benchmark Results

License

MIT