Example 9 — Non-Convex Enclosure: an L-Shaped Duct

Last updated

30 September 2026

The reentrant corner of an L-shaped duct shadows one arm from the other. View factors have no occlusion test, so this geometry requires ray tracing.

Three unit squares in cross-section, extruded to unit height. The z = 0 faces are held at 1000 K, the z = 1 faces at 0 K, and the eight side walls are in radiative equilibrium. All surfaces black.

Step 1: Geometry and boundary conditions

The L-shaped caps are hexagons, split into three quadrilaterals each. The side walls follow the boundary cycle of the cross-section.

using RayTraceHeatTransfer, GLMakie, StatsBase

xy = [0.0 0.0; 1.0 0.0; 2.0 0.0; 2.0 1.0;
      1.0 1.0; 0.0 1.0; 1.0 2.0; 0.0 2.0]
points = vcat(hcat(xy, zeros(8)), hcat(xy, ones(8)))   # z = 0 then z = 1

faces = [ 1  2  5  6;  2  3  4  5;  6  5  7  8;    # 1–3   z = 0 caps
          9 10 13 14; 10 11 12 13; 14 13 15 16;    # 4–6   z = 1 caps
          1  2 10  9;  2  3 11 10;  3  4 12 11;    # 7–9   sides, 9 = x-arm end
          4  5 13 12;  5  7 15 13;  7  8 16 15;    # 10–12 sides, 12 = y-arm end
          8  6 14 16;  6  1  9 14]                 # 13–14 sides

Ndim = 11
domainL = RayTracingDomain3D_surfaces(points, faces, Ndim,
              zeros(14), [fill(1000.0, 3); zeros(3); fill(-1.0, 8)], ones(14));

true

Winding does not matter — the constructor orients the faces and fixes the sign from the enclosed volume. Holes and non-manifold edges are rejected here.

Step 2: Trace, smooth and solve

In an interactive Julia session the scene can be zoomed into the duct and the reentrant corner inspected from within by hovering; on this page the figure is a static image.

domainL(10^8; verbose = false)     # total rays, divided across the 1694 elements
smooth!(domainL; verbose = false)
solveEquilibrium!(domainL, domainL.F_smooth; verbose = false)

# Zoom into the duct and inspect the reentrant corner from within:
fig  = Figure(size = (1200, 700))
ax   = LScene(fig[1, 1], scenekw = (camera = cam3d!, show_axis = true))
info = Label(fig[1, 2], ""; tellheight = false, tellwidth = true,
             halign = :left, justification = :left,
             fontsize = 14, font = "DejaVu Sans Mono") # with info as in Example 6
colsize!(fig.layout, 2, Relative(0.5))
plotMesh(ax, domainL; field = :T, inspect = true, label = info)
DataInspector(fig)
fig

Rays are traced to first intersection only; reflections are handled by the solver, so F_raw is geometry alone. Smoothing starts from a reciprocity defect of order 10⁻² rather than Example 6’s 10⁻¹³ — sampling breaks reciprocity at the noise level, not at roundoff — and reaches 10⁻¹⁵ either way.

The trace uses all threads by default, and the result does not depend on how many: it is fixed by the geometry, the ray count and seeds, a single integer selecting the realisation. Runs with different integers are statistically independent (see Sampling and reproducibility).

A ray either reaches a facet or it does not, so occlusion appears as exact structural zeros. The analytical method returns a substantial value for the same pair, having no notion of what lies between two polygons. Analytical view factors are exact and converge for free but cannot see around corners; ray tracing pays for every digit and has no geometric restriction. Example 6 shows the two agreeing on a cube, which is what licenses trusting the tracer here.

Rays are traced against a bounding volume hierarchy built with the binned surface area heuristic of Wald (2007), using the ray–triangle test of Möller and Trumbore (1997). This is what makes occlusion affordable: the cost of finding a ray’s first intersection grows logarithmically rather than linearly in the number of surface elements.

References

Möller, T. and Trumbore, B. (1997). Fast, minimum storage ray-triangle intersection. Journal of Graphics Tools, 2(1), 21–28.

Wald, I. (2007). On fast construction of SAH-based bounding volume hierarchies. Proceedings of the 2007 IEEE Symposium on Interactive Ray Tracing, 33–40.

--- title: "Example 9 — Non-Convex Enclosure: an L-Shaped Duct" execute: warning: false --- The reentrant corner of an L-shaped duct shadows one arm from the other. View factors have no occlusion test, so this geometry requires ray tracing. Three unit squares in cross-section, extruded to unit height. The *z* = 0 faces are held at 1000 K, the *z* = 1 faces at 0 K, and the eight side walls are in radiative equilibrium. All surfaces black. ## Step 1: Geometry and boundary conditions The L-shaped caps are hexagons, split into three quadrilaterals each. The side walls follow the boundary cycle of the cross-section. ```{julia} using RayTraceHeatTransfer, GLMakie, StatsBase xy = [0.0 0.0; 1.0 0.0; 2.0 0.0; 2.0 1.0; 1.0 1.0; 0.0 1.0; 1.0 2.0; 0.0 2.0] points = vcat(hcat(xy, zeros(8)), hcat(xy, ones(8))) # z = 0 then z = 1 faces = [ 1 2 5 6; 2 3 4 5; 6 5 7 8; # 1–3 z = 0 caps 9 10 13 14; 10 11 12 13; 14 13 15 16; # 4–6 z = 1 caps 1 2 10 9; 2 3 11 10; 3 4 12 11; # 7–9 sides, 9 = x-arm end 4 5 13 12; 5 7 15 13; 7 8 16 15; # 10–12 sides, 12 = y-arm end 8 6 14 16; 6 1 9 14] # 13–14 sides Ndim = 11 domainL = RayTracingDomain3D_surfaces(points, faces, Ndim, zeros(14), [fill(1000.0, 3); zeros(3); fill(-1.0, 8)], ones(14)); ``` ```{julia} #| echo: false Makie.inline!(true) # draw figures into the page rather than a window ``` Winding does not matter — the constructor orients the faces and fixes the sign from the enclosed volume. Holes and non-manifold edges are rejected here. ## Step 2: Trace, smooth and solve In an interactive Julia session the scene can be zoomed into the duct and the reentrant corner inspected from within by hovering; on this page the figure is a static image. ```{julia} domainL(10^8; verbose = false) # total rays, divided across the 1694 elements smooth!(domainL; verbose = false) solveEquilibrium!(domainL, domainL.F_smooth; verbose = false) # Zoom into the duct and inspect the reentrant corner from within: fig = Figure(size = (1200, 700)) ax = LScene(fig[1, 1], scenekw = (camera = cam3d!, show_axis = true)) info = Label(fig[1, 2], ""; tellheight = false, tellwidth = true, halign = :left, justification = :left, fontsize = 14, font = "DejaVu Sans Mono") # with info as in Example 6 colsize!(fig.layout, 2, Relative(0.5)) plotMesh(ax, domainL; field = :T, inspect = true, label = info) DataInspector(fig) fig ``` Rays are traced to first intersection only; reflections are handled by the solver, so `F_raw` is geometry alone. Smoothing starts from a reciprocity defect of order 10⁻² rather than [Example 6](example-06.qmd)'s 10⁻¹³ — sampling breaks reciprocity at the noise level, not at roundoff — and reaches 10⁻¹⁵ either way. The trace uses all threads by default, and the result does not depend on how many: it is fixed by the geometry, the ray count and `seeds`, a single integer selecting the realisation. Runs with different integers are statistically independent (see [Sampling and reproducibility](https://github.com/NikoBiele/RayTraceHeatTransfer.jl#sampling-and-reproducibility)). A ray either reaches a facet or it does not, so occlusion appears as exact structural zeros. The analytical method returns a substantial value for the same pair, having no notion of what lies between two polygons. Analytical view factors are exact and converge for free but cannot see around corners; ray tracing pays for every digit and has no geometric restriction. [Example 6](example-06.qmd) shows the two agreeing on a cube, which is what licenses trusting the tracer here. Rays are traced against a bounding volume hierarchy built with the binned surface area heuristic of Wald (2007), using the ray–triangle test of Möller and Trumbore (1997). This is what makes occlusion affordable: the cost of finding a ray's first intersection grows logarithmically rather than linearly in the number of surface elements. ## References Möller, T. and Trumbore, B. (1997). Fast, minimum storage ray-triangle intersection. *Journal of Graphics Tools*, 2(1), 21–28. Wald, I. (2007). On fast construction of SAH-based bounding volume hierarchies. *Proceedings of the 2007 IEEE Symposium on Interactive Ray Tracing*, 33–40.