V-HC-04 — two-layer slab, k-ratio 10.0:1
Tags: heat_conduction, laminate, harmonic_mean, steady
References:
- series thermal-resistance network (exact)
- FDS Verification Guide
insulated_steel_plate/ht_pile
Problem statement
FDS insulated_steel_plate / ht_pile analog. Two inert solids layered via composition step ICs; Dirichlet T at both faces; steady exact solution is the series-resistance network:
q = ΔT / (d₁/k₁ + d₂/k₂), Tinterface = Tbot + q·d₁/k₁, piecewise linear
What is verified: the distance-weighted harmonic-mean face conductivity is NODALLY EXACT for piecewise-constant k with the jump on a face — the steady cell-center temperatures and the boundary fluxes must match the resistance network to integrator precision, for k-ratios 10:1 and 100:1.
Off-face variant (c): composition step ICs are SAMPLED AT CELL CENTERS, so a prescribed interface that is not on a face quantizes to the nearest face — there are no mixed cells. Measured 2026-07-06: the numerics match the QUANTIZED-interface network to 1e-11 (i.e. the scheme is still nodally exact), and the entire user-facing error is the true-vs-quantized network gap (1.64% in flux, 1.47 K in profile at Δz = 0.5 mm) — pinned here against its a-priori Δz/2 sensitivity bound.
No convergence spec: (a)/(b) are nodally exact (error is integrator-, not mesh-limited); (c) is quantization-dominated and non-monotone in n.
V-HC-04a — two-layer slab, k-ratio 10.0:1
Quantities of interest (n = 40)
| QoI | value | exact | error | tolerance | within tol | provenance |
|---|---|---|---|---|---|---|
| L∞ steady profile vs (quantized) network | 1.518e-11 | — | 1.518e-11 | 1e-08 | yes | harmonic-mean face k is NODALLY EXACT for face-aligned jumps; k-ratio 10.0:1 |
| steady bottom-face flux vs network | 909.1 | 909.1 | 3.336e-11 | 1e-08 | yes | both faces must carry the exact series flux |
| steady top-face flux vs network | 909.1 | 909.1 | 6.595e-13 | 1e-08 | yes | both faces must carry the exact series flux |
| steadiness (last half-interval drift) | 8.219e-10 | — | 8.219e-10 | 1e-06 | yes | tend = 10·L²/αmin |
Solver configuration
| setting | value |
|---|---|
| integrator | KenCarp4 (default) |
| abstol | 1.0e-10 |
| reltol | 1.0e-8 |
V-HC-04b — two-layer slab, k-ratio 100.0:1
Quantities of interest (n = 40)
| QoI | value | exact | error | tolerance | within tol | provenance |
|---|---|---|---|---|---|---|
| L∞ steady profile vs (quantized) network | 3.024e-11 | — | 3.024e-11 | 1e-08 | yes | harmonic-mean face k is NODALLY EXACT for face-aligned jumps; k-ratio 100.0:1 |
| steady bottom-face flux vs network | 99.01 | 99.01 | 4.082e-10 | 1e-08 | yes | both faces must carry the exact series flux |
| steady top-face flux vs network | 99.01 | 99.01 | 9.774e-14 | 1e-08 | yes | both faces must carry the exact series flux |
| steadiness (last half-interval drift) | 3.112e-09 | — | 3.112e-09 | 1e-06 | yes | tend = 10·L²/αmin |
Solver configuration
| setting | value |
|---|---|
| integrator | KenCarp4 (default) |
| abstol | 1.0e-10 |
| reltol | 1.0e-8 |
V-HC-04c — two-layer slab, k-ratio 10.0:1 (off-face interface)
Quantities of interest (n = 40)
| QoI | value | exact | error | tolerance | within tol | provenance |
|---|---|---|---|---|---|---|
| L∞ steady profile vs (quantized) network | 1.518e-11 | — | 1.518e-11 | 1e-08 | yes | harmonic-mean face k is NODALLY EXACT for face-aligned jumps; k-ratio 10.0:1 |
| steady bottom-face flux vs network | 909.1 | 909.1 | 3.336e-11 | 1e-08 | yes | both faces must carry the exact series flux |
| steady top-face flux vs network | 909.1 | 909.1 | 6.595e-13 | 1e-08 | yes | both faces must carry the exact series flux |
| interface-quantization smearing (documented) | 0.01636 | 0 | 0.01636 | 0.02079 | yes | THE laminate-off-mesh error users see: flux off by 1.64% at Δz = 0.0005 m |
| steadiness (last half-interval drift) | 8.219e-10 | — | 8.219e-10 | 1e-06 | yes | tend = 10·L²/αmin |
Solver configuration
| setting | value |
|---|---|
| integrator | KenCarp4 (default) |
| abstol | 1.0e-10 |
| reltol | 1.0e-8 |