Heat-load estimators¶

This page defines how SIMPLE’s post-processing turns lost markers into deposited power and heat flux, on flux-surface bins or on STL wall triangles, and what limits the accuracy of these estimates. Usage: Wall heat loads.

Marker power and energy¶

For D-T, \(P_\alpha = P_\text{fus}/5\). Marker \(i\) represents the share \(w_i/\sum_{k=1}^N w_k\) of the source alpha power, with relative weights \(w_i \ge 0\); equal weights give \(P_\alpha/N\). Equal weights represent the physical source only when births are sampled in proportion to it (startmode = 7, Fusion-weighted alpha births).

All markers start with the reference energy \(E_0 = 3.5\,\text{MeV}/\)facE_al, \(v_0=\sqrt{2E_0/m}\). At the exit the marker carries

\[ \frac{E_i}{E_0} = p_i^2, \qquad p_i = \frac{v_i}{v_0} = \texttt{zend(4)} . \]

Without collisions \(p_i = 1\). With collisions the energy given to the background before the exit is not deposited on the wall; \(p_i < 1\) on average, but energy diffusion can raise individual markers above \(p_i = 1\). Mixed birth energies need a separate source convention and are not handled.

The particle and energy loss fractions and the lost power are

\[ f_\text{particle} = \frac{n_\text{lost}}{N}, \qquad f_\text{energy} = \frac{\sum_{i\in\text{lost}} w_i p_i^2}{\sum_{k} w_k}, \qquad P_\text{lost} = f_\text{energy} P_\alpha . \]

Flux on flux-surface bins¶

Lost markers that crossed the LCFS (exit code 1) are binned by their exit angles \((\vartheta_i, \varphi_i)\) = zend(2:3). With equal weights

\[ q_\text{bin} = \frac{P_\alpha}{N A_\text{bin}} \sum_{i \in \text{bin}} p_i^2 . \]

The bin area comes from the outermost chartmap surface \(\mathbf r(\vartheta,\zeta)\) (the LCFS for a chartmap written by vmec_to_chartmap.x), stored for one field period and repeated \(n_\text{fp}\) times:

\[ A_\text{bin} = \iint_\text{bin} \left|\frac{\partial \mathbf r}{\partial \vartheta}\times \frac{\partial \mathbf r}{\partial \zeta}\right| \mathrm d\vartheta\,\mathrm d\zeta , \qquad A = n_\text{fp}\int_0^{2\pi}\!\!\int_0^{2\pi/n_\text{fp}} \left|\partial_\vartheta \mathbf r\times\partial_\zeta \mathbf r\right| \mathrm d\zeta\,\mathrm d\vartheta . \]

The integral is a sum over chartmap grid cells whose nodes lie in the bin, so a bin needs grid points: use at most ntheta poloidal and nzeta\(\times n_\text{fp}\) toroidal bins of the chartmap, otherwise bins get zero area. Bin angles and chartmap angles must be the same coordinates (VMEC angles for vmec_to_chartmap.x). Implementation: compute_bin_areas_from_chartmap in python/pysimple/engineering.py.

Load on STL triangles¶

A marker with wall_hit == 1 deposits on its facet \(j\) = wall_hit_triangle. For facet area \(A_j\) (m²)

\[ P_j = P_\alpha\,\frac{\sum_{i:\,\text{hit}(i)=j} w_i\, p_i^2}{\sum_{k=1}^{N} w_k}, \qquad q_j = \frac{P_j}{A_j}. \]

The denominator runs over all source markers, including confined markers and losses that did not reach the STL. \(\sum_j P_j\) is the deposited part of \(P_\alpha\). No incidence cosine is applied: the power lands on the actual facet area, not a projected one. Weights come from marker_weights, else from a marker_weight(particle) variable in results.nc (SIMPLE does not write one), else all ones. Facets of zero area keep their index, receive no power and add no area; a recorded hit on one is an error. Implementation: TriangleHeatLoad.from_netcdf in python/pysimple/wall_heat_load.py.

Metrics of TriangleHeatLoad.summary:

Metric	Definition
`peak_W_m2`	\(\max_j q_j\)
`p95_W_m2`	area-weighted 95th percentile over all facets, unhit included: sort \(q_j\), take the first whose cumulative area reaches 95 % of the total
`wetted_area_m2(t)`	\(\sum A_j\) over facets with \(q_j > t\) (default \(t = 0\))
`incidence_distribution(bins=18)`	hit count and deposited power in angle bins on \([0, \pi/2]\)

The incidence angle is \(\arccos|\hat{\mathbf s}\cdot\hat{\mathbf n}|\) between the guiding-centre step chord \(\hat{\mathbf s}\) and the facet normal \(\hat{\mathbf n}\) (0 is normal incidence), not a gyrophase-resolved velocity. The normal follows the STL vertex winding and need not point outward. The loss time is interpolated linearly along the intersecting chord (locate_wall_segment in src/simple_main.f90).

Accuracy¶

Counting noise. A bin or facet with \(k\) hits has relative error about \(1/\sqrt{k}\). No smoothing is applied; small facets with few hits produce large peaks. Converge peak values in marker number.
Guiding centre, not impact point. Positions are guiding-centre positions. The particle hits up to one gyroradius away, \(\rho_\alpha = m_\alpha c\, v_\perp/(Z_\alpha e B) \le 5.4\) cm for 3.5 MeV alphas at 5 T. Fine wall features and peak locations below this scale are not resolved.
Hit energy. With collisions, the hit keeps the momentum of the kicks applied before the intersecting microstep; the energy has the time resolution of the microstep.

References¶

C. G. Albert et al., Symplectic methods for optimization of wall heat loads by fusion alpha particles in stellarator reactors, Nucl. Mater. Energy 48, 102200 (2026), doi:10.1016/j.nme.2026.102200.