1024 laminar heating solutions on the heat shield of a parametric entry capsule, from a Newtonian panel method: modified Newtonian pressure, Fay–Riddell stagnation heating and the Lees laminar distribution. Four shape parameters, including the capsule's real size from a Varda-like 0.8 m to beyond Orion's 5 m, vary together with Mach number and altitude. Every capsule's wall mesh comes from one TopoLink family topology, so node i is the same point of the heat shield on every capsule. The wall fields of all runs therefore stack into one array, ready to fit a surrogate without interpolating between meshes.

The whole dataset is one Gamut file. It holds the run table, the mesh and every field, all with units, and it is readable from Python, MATLAB, Julia, C++, C, Rust and the command line.

This is a demonstration dataset. It shows a workflow: parametric geometry, matched meshes, a batch of solutions and one self-describing file. The model is a fast low-fidelity one with the known errors of Newtonian methods; see Limitations.

Above: 180 capsules along a loop through the family, from the Varda-like shape through the Gemini-like and Orion-like ones to the Mars-entry-like one, each at its real size. Every frame has its own CAD, TopoLink mesh and solution; nothing is interpolated between frames. The mesh panel marks the family's block layout in orange. The loop is held at Mach 10 and 60.6 km and is not part of the dataset.

At a glance


Runs	1024 sampled shapes, each at its own flight condition
Inputs	diameter, nose radius, forebody angle and shoulder radius (shape); Mach number and altitude (flight)
Wall fields	heat flux, pressure and arc length from the stagnation point, on 2929 nodes and 2880 quads of the full heat shield
Scalars per run	heat load; stagnation, peak and mean heat flux; stagnation velocity gradient; freestream state
Ranges	stagnation heat flux 2.49–1884 kW/m², heat load 0.0029–8.7 MW
Model	modified Newtonian pressure, laminar similarity stagnation heating, Lees distribution; ideal-gas air
File	`vehicle_capsule_newtonian.gamut`, 55.1 MB: runs, inputs, results, mesh and fields, all with units

Inputs

Name	Range	Units	Meaning
`diameter`	0.8–5.5	m	maximum diameter D, sampled log-uniform
`nose_radius`	0.25–1.6	× D	radius of the spherical nose cap
`forebody_angle`	55–75	degree	half-angle of the cone between the nose cap and the shoulder
`shoulder_radius`	0.025–0.075	× D	radius of the toroidal shoulder
`mach`	6–15	1	freestream Mach number
`altitude`	40–80	km	freestream from the 1976 US Standard Atmosphere

Held fixed for every run: zero angle of attack, a 300 K isothermal wall, and ideal-gas air (γ = 1.4, R = 287.05 J/(kg K), Sutherland viscosity, Pr = 0.72). The capsule family also varies the backshell, but the domain is the heat shield alone, so the backshell plays no part.

The inputs come from a Latin hypercube over all six, optimised for low centred discrepancy with Proximant's sampler, so each input's range is split into 1024 equal strata with one design point in each. The family builds every forebody from a spherical nose cap, a cone and a toroidal shoulder, and the cone must be at least 1 % of the diameter long. A blunt nose on a shallow cone would leave no cone, so the largest nose radius falls from 1.6 D at 75° to about 0.8 D at 55°. The nose radius is therefore drawn as a fraction of the largest buildable at each run's forebody angle and shoulder radius, and every run is buildable. The empty corner of the nose radius–forebody angle panel is that limit. Shapes at the limit are smooth sphere-cap heat shields, like Apollo's and Orion's.

The file

vehicle_capsule_newtonian.gamut is the whole dataset. It holds the run table (labels, inputs and results, each with units, and the inputs with their bounds), the wall coordinates of each run, the quad connectivity shared by every run, the fields at the nodes, and the fixed constants. gamut info describes it:

vehicle_capsule_newtonian.gamut: shared mesh, 1,024 runs
  mesh        2,929 nodes, 2,880 cells, geometry per member
  parameters  altitude [km] 40.027 to 79.969, diameter [m] 0.8008 to 5.4921, forebody_angle [degree] 55.0022 to 74.9898, mach [1] 6.006 to 14.995, nose_radius [1] 0.25095 to 1.57933, shoulder_radius [1] 0.0250237 to 0.0749567
  labels      kind (categorical): 1024 sample
  labels      member (group): 1024 names, 1 run each: capsule_0000, capsule_0001, capsule_0002, ...
  results     freestream_density [kg m-3], freestream_pressure [Pa], freestream_temperature [K], freestream_velocity [m s-1], heat_load [W], mean_heat_flux [W m-2], nose_radius_fraction [1], peak_heat_flux [W m-2], peak_heat_flux_arc_length [m], stagnation_heat_flux [W m-2], stagnation_velocity_gradient [s-1]
  fields      arc_length [m] at nodes; heat_flux [W m-2] at nodes; pressure [Pa] at nodes
  constants   angle_of_attack = 0.0 degree, gas_constant = 287.05 J kg-1 K-1, prandtl_number = 0.72 1, ratio_of_specific_heats = 1.4 1, wall_temperature = 300.0 K
  file        55.1 MB

Coordinates are in metres at each capsule's real size. x runs along the capsule axis, downstream from the stagnation point. Each run was computed on the half heat shield z ≥ 0 and mirrored across z = 0. The nodes on the plane z = 0 are shared and the mirror's nodes come after the half's, so the first 1501 nodes are the half that was computed (q[:, :1501] in the Python example below).

Because every capsule has the same nodes, a field of the whole study is one array of runs by nodes, and each column of it is one point of the heat shield:

Quick start

pip install bijective-gamut
hf download bijective-systems/capsule-aeroheating --repo-type dataset --local-dir capsule
gamut info capsule/vehicle_capsule_newtonian.gamut

In Python:

import gamut

with gamut.open("capsule/vehicle_capsule_newtonian.gamut") as study:
    runs = study.table()                    # inputs, results and labels, one entry per run
    q = study.get_field("heat_flux")        # (runs, nodes), W m-2
    xyz = study.get_points(0)               # wall nodes of the first run, m
    cells = study.get_cells()               # the shared quads

From the command line, without writing code:

# a table of chosen inputs and results, as CSV
gamut get capsule/vehicle_capsule_newtonian.gamut member diameter mach altitude heat_load -o table.csv

# one .vtu per run, with every field, for ParaView
pip install "bijective-gamut[vtk]"
gamut-vtk export capsule/vehicle_capsule_newtonian.gamut capsule_vtk/

Gamut has libraries for MATLAB, Julia, C++, C and Rust as well; see the documentation.

How it was made

Geometry. Each capsule's CAD comes from the vehicle_capsule parametric family, built with OpenCASCADE and written as STEP at the capsule's real diameter.
Surface mesh. TopoLink maps the family's block topology onto each capsule's CAD and smooths it, with 24 cells from the central block to the shoulder. Every capsule gets the same nodes and connectivity, so the fields correspond node by node. The heat shield ends partway round the shoulder.
Pressure. Modified Newtonian: the stagnation pressure behind a normal shock times the square of the cosine between the flow and the wall normal.
Heating. At the stagnation point, the exact laminar boundary-layer similarity solution for ideal-gas air (the ideal-gas form of Fay–Riddell), at the Newtonian velocity gradient of the nose radius. Elsewhere, the Lees laminar distribution along the meridian, with the edge velocity from an isentropic expansion to the local Newtonian pressure. At zero incidence every streamline is a meridian.
Packaging. Each run's half wall and fields are mirrored across the symmetry plane to the full heat shield. All runs go into one Gamut file, checked with gamut validate.

A run takes about 20 s with four running at once on a desktop CPU (AMD Ryzen 5 3600X), almost all of it TopoLink meshing.

Limitations

The model was checked against laminar SU2 Navier–Stokes solutions on the same wall mesh for three shapes at their real sizes:

Small nose, sphere–cone (Varda-like 0.9 m at Mach 15, 40 km). Stagnation heat flux within 2 % of SU2 and heat load 2 % low.
Large-radius heat shield (Orion-like 5.02 m at Mach 15, 40 km). Heat load 14 % low and stagnation heat flux 13 % low. The shoulder speeds up the flow across the whole face: SU2's stagnation velocity gradient matches an effective nose radius of 4.5 m, not the geometric 6.0 m. Newtonian methods miss this unless they model the effective radius.
70° cone (Mars-entry-like 4.5 m at Mach 6, 80 km). Stagnation heat flux 26 % high and heat load 12 % low. The sonic point sits at the shoulder, so the nose heats as if its radius were larger, and the cone's pressure is not Newtonian.
The shoulder heating peak is under-predicted on every shape.
Ideal gas. There is no chemistry and no vibrational excitation. At the top of the Mach range, real air dissociates and the heat flux would differ.
Laminar. There is no transition and no turbulence.
Heat shield only, at zero incidence. There is no afterbody and no wake, and the two halves of each heat shield carry the same information.

Licence and citation

The data and figures are licensed under CC BY 4.0.

@misc{bijective_capsule_aeroheating_2026,
  title        = {Capsule forebody aeroheating, engineering model (demo)},
  author       = {{Bijective Systems}},
  year         = {2026},
  howpublished = {\url{https://huggingface.co/datasets/bijective-systems/capsule-aeroheating}}
}

About

Made by Bijective Systems with TopoLink (meshes for parametric geometry families) and Gamut (an open file format for parametric simulation studies); the CFD check used SU2. Proximant, a Bijective Systems product, fits surrogate models to studies like this one.

Exported 2026-11-04.

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