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Physics-Informed Prognostics of Fatigue Failure in Additively Manufactured Rocket-Nozzle Cooling Channels: Sparse-Sensor Observability, Bayesian Calibration, and Uncertainty Quantification

This study proposes a physics-informed Bayesian digital twin that infers manufacturing defects and predicts fatigue life in additively manufactured rocket-nozzle cooling channels using sparse sensor data, demonstrating superior robustness to out-of-distribution process variations compared to data-driven regressors while highlighting the critical need for additional non-destructive testing to detect unobservable porosity.

Original authors: Viraj Gajera

Published 2026-10-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Viraj Gajera

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Rocket engines are the most powerful machines humans build, yet they are also among the most fragile. In a reusable liquid-fuel engine, the nozzle—the part that directs the scorching exhaust into space—must survive temperatures hot enough to melt steel while being cooled by fuel rushing through tiny internal channels. This cooling system, known as regenerative cooling, is a delicate balance. If the metal wall between the burning gas and the cooling fuel is too thin, or if it contains hidden flaws from its creation, the engine can crack and fail after only a few uses. For decades, engineers have relied on average statistics to decide when to retire these parts, assuming that every engine in a fleet is roughly the same. But as manufacturing techniques evolve to print these complex engines layer by layer, the reality has become more complicated. The very process that allows for intricate designs also introduces tiny, unpredictable variations in the metal, such as microscopic pores or rough surfaces, which can drastically shorten an engine's life. The critical question for the future of spaceflight is whether we can detect these specific weaknesses in a single engine before it fails, using only the limited sensors available during a test.

A researcher at the New York Institute of Technology has tackled this challenge by creating a sophisticated computer simulation to test how well we can predict the lifespan of these printed rocket nozzles. The study focuses on a specific type of engine nozzle made from a copper alloy, which is printed using a laser that fuses metal powder. While this manufacturing method is revolutionary, it leaves behind subtle defects like internal roughness and tiny holes. The researcher built a "digital twin," a virtual copy of a real engine nozzle that learns from data. This twin is designed to take two simple measurements taken during a single engine firing—the rise in temperature of the cooling fuel and the drop in pressure as it flows through the channels—and work backward to guess the internal condition of the metal wall. The goal was to see if these two global signals could reveal the hidden state of the engine well enough to predict when it would break.

The simulation tested three different scenarios using virtual fleets of three hundred engines each. In the first scenario, the engines had typical manufacturing variations. In the second, the engines were made with a rougher surface finish, a defect that the sensors could detect. In the third, the engines were built with a batch of powder containing larger internal pores, a defect that the sensors could not see. The digital twin was pitted against other prediction methods, including complex statistical models trained on data from engines that had been run until they failed. The results showed that the digital twin was remarkably effective at identifying engines with rough surfaces or incorrect wall thicknesses. After just three simulated firings, the system could correctly flag engines that were likely to fail early with a high degree of accuracy. It achieved this by inferring the average thickness of the wall and the roughness of the channel, which directly influence how the engine handles heat and pressure.

However, the study also revealed a hard limit to what sensors can tell us. When the engines contained hidden internal pores, the sensors were completely blind. The temperature and pressure readings looked normal, even though the engine was dangerously flawed. In these cases, the digital twin and all other prediction methods failed to provide a reliable estimate of the engine's life, often giving a false sense of security. The only way to fix this blind spot in the simulation was to add a direct inspection measurement, specifically a scan that could see the size of the internal pores. When this extra data was fed into the digital twin, its predictions became accurate again. This finding suggests that while sensors are excellent at catching manufacturing errors that change the overall shape or flow of the engine, they cannot see defects that are buried deep inside the metal.

Another crucial discovery was that the digital twin remained reliable even when the engines were made with rougher surfaces than the system had ever seen before. Other prediction methods, which rely purely on patterns found in past data, struggled when faced with these new conditions. They became less accurate and less confident in their estimates. The digital twin, because it is built on the fundamental laws of physics rather than just historical patterns, was able to adapt and continue providing calibrated, trustworthy predictions. This highlights a key advantage of using physics-based models: they do not break down as easily when the manufacturing process shifts slightly outside the norm.

The research also emphasized the importance of accounting for the difference between the simplified computer model and the complex reality of the engine. Without a specific correction factor to bridge this gap, the digital twin became overly confident in its wrong answers, narrowing its predictions to a precise but incorrect value. By including a term to represent this uncertainty, the system maintained a realistic range of possibilities, ensuring that it did not miss potential failures. The study concludes that while we can use simple sensor data to identify many manufacturing defects and predict the life of a rocket nozzle, we cannot rely on sensors alone. To truly ensure safety, especially against hidden internal flaws, we must combine sensor data with direct inspection methods. This hybrid approach offers the best path forward for managing the lifespan of the next generation of reusable rocket engines, moving away from guessing based on averages and toward knowing the specific condition of each individual machine.

Technical Summary: Physics-Informed Prognostics of Fatigue Failure in Additively Manufactured Rocket-Nozzle Cooling Channels

Problem Statement
Laser powder bed fusion (L-PBF) enables the fabrication of complex, integral cooling channels in copper-alloy rocket thrust chambers (e.g., GRCop-42). However, this manufacturing process introduces defects such as wall-thickness deviations, internal roughness, and porosity, which significantly reduce thermomechanical fatigue life. The core challenge addressed is determining whether sparse, global sensors available on a test stand or in flight (specifically coolant temperature rise $\Delta T$ and pressure drop $\Delta p$ ) can identify these defects to enable part-specific life management. Current literature lacks a joint analysis of defect identifiability, the value of part-specific data beyond fleet averages, uncertainty calibration, and the failure modes of prognostic methods under distribution shifts.

Methodology
The author developed a reproducible in-silico benchmark to test a physics-informed Bayesian digital twin against data-driven regressors.

Reference Hardware & Physics: The study models a generic LOX/CH4 rocket throat liner. The "truth" model is a high-fidelity 2-D finite-volume conduction solver with grid-converged error (0.02%). The digital twin utilizes a simplified 1-D series resistance model with rib-fin enhancement, corrected via a log-linear discrepancy term fitted to a small set of high-fidelity runs.
Fatigue & Damage Model: Life prediction follows a constrained-liner idealization. Damage accumulation uses Miner's rule with low-cycle fatigue (LCF) life derived from Manson's universal slopes, modified by Murakami's $\sqrt{area}$ scaling for porosity and an Arola–Williams concentration factor for surface roughness. Ratchet thinning is modeled as proportional to plastic strain range. Failure is defined by crack initiation or wall rupture ( $t \leq 0.35$ mm).
Experimental Design: Four fleets of 300 parts each were generated:
1. Training/In-Distribution (ID): Standard manufacturing variability.
2. OOD-Rough: Observable shift (increased median roughness $R_a$ ).
3. OOD-Pore: Unobservable shift (increased median pore size $a$ ).
4. Oracle: Knows all defects except life scatter.
Prognostic Methods:
- Bayesian Twin: Infers the posterior distribution of global state ( $\bar{t}, R_a$ ) from sparse sensor data ( $\Delta T, \Delta p$ ) per firing, propagating uncertainty through the damage model. It explicitly includes a model-discrepancy term (Kennedy–O'Hagan) to account for 1-D model bias.
- Benchmarks: Gaussian Process (GP) regression, Quantile Gradient Boosting (QGB), a Hybrid (QGB fed twin features), and Fleet Empirical Quantiles.
Metrics: Continuous Ranked Probability Score (CRPS) for life prediction, 90% interval coverage (calibration), and AUC/Brier scores for early failure detection ( $N_f < B_{10}$ ).

Key Contributions

Identifiability Framework: The study quantifies that sparse global sensors can identify global manufacturing states (mean wall thickness and roughness) but are fundamentally blind to local defects (local thinning and pore size).
Discrepancy-Aware Calibration: The author demonstrates that a physics-based twin requires an explicit model-discrepancy term. Without it, parameter posteriors become confidently wrong (coverage drops to 36%) as data accumulates, even if life predictions appear acceptable due to aleatory scatter masking the bias.
Distribution Shift Robustness: The twin maintains calibrated uncertainty intervals under observable shifts (roughness) where data-driven regressors degrade. However, under unobservable shifts (porosity), sensor-only methods fail completely unless inspection data (CT scans) are integrated.
Open Benchmark: The paper provides a fully reproducible, open-source in-silico benchmark including code, synthetic fleets, and verification records for fatigue prognosis in additive manufacturing.

Results

In-Distribution Performance: All sensor-based methods reduced the CRPS of log-life from 0.086 (fleet quantiles) to 0.075–0.077 dex. The Bayesian twin achieved 92% coverage on 90% intervals, outperforming QGB (83%) and GP (89%).
Early Failure Detection: After three firings, the twin identified out-of-specification parts with an AUC of 0.94.
Observable Shifts (Roughness): When roughness shifted outside the training range, regressors lost calibration (coverage dropped to 0.78–0.84) and CRPS increased by 0.017–0.025 dex. The twin maintained 91% coverage, supporting the hypothesis that physics-based models extrapolate better than pure data-driven models.
Unobservable Shifts (Porosity): For a porous powder lot, all sensor-only methods failed (coverage $\leq 0.48$ ). The twin was the worst performer (CRPS 0.221) because it relied on an incorrect prior for pore size. However, adding a single CT measurement restored the twin's coverage to 0.88 and CRPS to 0.076, outperforming CT-augmented regressors.
Sensor Value: Sensors recovered approximately one-third of the gap between fleet-average uncertainty and the oracle (ideal) uncertainty. Adding CT data recovered approximately three-quarters.

Significance and Claims
The paper claims a "qualified yes" to the question of whether flight-like instrumentation can reveal defects before failure. The findings suggest that:

Global sensors are sufficient to flag parts with observable defects (roughness, thickness) that lead to early failure, but they cannot forecast failures driven by local, invisible defects (porosity).
Physics-informed twins offer superior calibration under distribution shifts compared to data-driven regressors, provided model discrepancy is explicitly quantified.
To achieve reliable prognostics for L-PBF parts, inspection data (e.g., CT) is required to constrain unobservable variables; better algorithms applied solely to global signals cannot overcome the lack of observability.
The primary vulnerability of the twin lies in the damage law itself (e.g., ductility assumptions), making material LCF data for L-PBF alloys the highest-value validation target.

The author explicitly states that these results are simulation-based and await validation with hot-fire data. They do not claim to have solved the damage law validity for L-PBF materials but have established a rigorous framework for testing prognostic hypotheses.

Technical Summary: Physics-Informed Prognostics of Fatigue Failure in Additively Manufactured Rocket-Nozzle Cooling Channels

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