Algorithms

This page gives a brief overview of the computational methods used inside tracketch. For full API details see API Reference.

Simulation pipeline

A TrackSimulator runs three stages:

1. Dose map – For each depth step along the ion path, the radial dose distribution (RDD) is evaluated using a model from libamtrack (e.g. Cucinotta). SRIM stopping-power tables provide LET and residual energy at each depth, so the RDD changes as the ion slows down.

2. Etch-rate map – The calibrated EtchRateModel converts local dose D to a local etch velocity V(D) via monotonic PCHIP interpolation in log-dose space. Undamaged material etches at the constant bulk rate V_bulk.

3. Arrival-time map – Starting from the detector surface (z = 0), the etchant front is propagated through the spatially varying speed field. This is a shortest-path / wavefront problem solved by one of the backends below.

Arrival-time solvers

The etch-rate map defines a speed at every grid point. The arrival time at each cell is the minimum travel time from the surface through any path, weighted by inverse speed:

\[T(\mathbf{x}) = \min_{\text{paths}} \int_0^{\mathbf{x}} \frac{\mathrm{d}s}{V(s)}\]

tracketch provides two solver families.

Dijkstra (recommended)

The 2-D grid is treated as a weighted graph. Each cell is a node connected to its neighbours; edge weights are the physical travel time across the cell boundary:

\[w_{ij} = \frac{|\mathbf{x}_i - \mathbf{x}_j|} {\tfrac{1}{2}(V_i + V_j)}\]

Because the radial grid is log-spaced (necessary to resolve narrow RDDs at small r), cell sizes vary by orders of magnitude. Dijkstra handles non-uniform spacing natively – no regridding required.

Connectivity. Higher connectivity reduces metrication error (the angular bias from a discrete grid):

• 8-connected (default) – king’s moves; ~2 % error; fastest.

• 16-connected – adds knight’s moves (+/-2, +/-1); ~1 % error.

• 32-connected – extended knight’s moves; ~0.5 % error.

Backends. The graph is built with Numba JIT and then solved with scipy.sparse.csgraph.dijkstra:

• dijkstra_numba – pure Python + Numba; always available.

• dijkstra_cpp – C++/pybind11 implementation of the graph builder; ~50-100x faster. Requires compilation (see Getting Started).

Fast Marching Method (FMM)

The scikit-fmm library solves the Eikonal equation on a uniform grid. Since the native radial grid is log-spaced, tracketch interpolates the speed map onto a uniform grid, runs FMM, then interpolates back.

FMM supports tilted tracks (theta_deg > 0), where the wavefront starts along a diagonal rather than a flat surface. Dijkstra does not yet support tilting.

Etch-rate model

EtchRateModel represents V(D) as a monotonic PCHIP (Piecewise Cubic Hermite Interpolating Polynomial) spline through a set of dose-velocity anchor points. Key design choices:

• Anchor velocities are stored as cumulative increments to enforce monotonicity by construction.

• A synthetic low-dose anchor at 10^-3 Gy is prepended so that the spline smoothly converges to V_bulk at low doses.

• Extrapolation beyond the highest anchor uses the last-segment slope (PCHIP mode) or clamps to the last value.

The model can be serialised to JSON and shared across simulations.