Speaker
Description
Background
NASA’s Dragonfly mission, which aims to deliver a rotorcraft to Saturn’s moon Titan, must withstand extreme convective and radiative heating during its entry into the moon’s dense nitrogen-methane atmosphere. Because of strongly radiating species, particularly the cyanogen (CN) radical, the forebody radiative heating can be as much as 50% of the total heat flux, necessitating the inclusion of radiation analysis when modeling the heating of the probe during its entry.
Engineering predictions of the heat flux to the spacecraft surface – both the forebody and afterbody – are subject to uncertainty resulting from unknown freestream methane content of Titan’s atmosphere, as well as uncertainties in the parameters used to model chemical kinetics, radiative emission, and solid conduction in the heat shield. These uncertainties carry major implications for the entry system reliability and so their effects must be quantified.
Methodology
For the reliability analysis of the forebody heat shield, the quantity of interest is the bondline temperature: an interior quantity, defined at the interface between the thermal protection system (TPS) and the underlying structure, whose exceedance of a critical threshold would compromise the integrity of the heat shield and the survival of the payload. Simulations of the forebody flow field are performed with US3D, a computational fluid dynamics (CFD) solver developed at the University of Minnesota. Radiation calculations are performed using the Multi-Fidelity Radiation Package (MURP), developed at the University of Illinois and the University of California, Irvine, and which is coupled to US3D using the PreCICE library. The bondline temperature is obtained by supplying the surface convective (US3D) and radiative (MURP) heating distributions to a one-dimensional solid thermal solver, with one calculation performed at every wall grid point through the thickness of the thermal protection system (TPS).
Within each US3D calculation, the grid is tailored – typically 2–4 times – in order to minimize numerical errors originating from poor alignment of cell faces with the bow shock. Coupling between US3D and MURP can be either one-way, with data flowing from US3D into MURP only once, at the end of the calculation, or two-way, in which US3D and MURP actively exchange data with one another throughout the calculation. Two-way coupling enables the US3D flowfield to respond to the flux of radiated power into or out of each cell.
Sensitivity and uncertainty quantification for the radiative environment requires propagating a large set of uncertain chemical-kinetic and CN electronic-excitation rate coefficients, together with the freestream methane content, through the flow–radiation model. Because a single high-fidelity (HF) US3D–MURP evaluation costs on the order of hundreds of CPU-hours, direct propagation over the hundreds of samples needed for variance-based sensitivity analysis is computationally prohibitive. To retain HF accuracy at a tractable cost, a bi-fidelity (BF) surrogate is constructed.
The BF method chosen is interpolative decomposition (ID) [1] due to the correlation obtained between the LF and HF models. A dense low-fidelity (LF) ensemble is generated using a coarse grid simulation of the forebody, which broadly covers the uncertain input space, with each sample running in approximately 10 CPU-hours. A pivoted QR factorization of the LF ensemble identifies a small set of maximally informative skeleton samples and an associated interpolation matrix; the US3D–MURP model is then evaluated only at these skeleton samples, and the full HF dataset is reconstructed by applying the LF-derived interpolation weights to the sparse HF evaluations. For the HF simulations, a finer computational grid with less than 2% error on radiative heating and 1% error on convective heating is used. Because the LF and HF radiance fields are strongly correlated and the data singular values decay rapidly, the QoI profiles are recovered to within roughly one percent in the Frobenius norm using only a handful of HF simulations, reducing the cost of the analysis by approximately two orders of magnitude relative to a full HF ensemble. The BF surrogate is then used to perform a Sobol analysis to identify the kinetic, excitation, and composition parameters that most strongly drive the bondline temperature variability, and the reliability assessment evaluates the heat shield against its thermal design margin.
Results
Before proceeding with sample generation, differences in forebody radiative heating between one- and two-way coupling are evaluated first. Results from ten sample points evaluated at the maximum convective heating trajectory point reveal a difference of approximately 1% between the radiative heating obtained from the two-way vs. one-way US3D–MURP coupling. As a result, both the high- and low-fidelity samples are generated using one-way radiation coupling. Future analyses will evaluate this assumption at the peak radiative heating trajectory point, where a non-negligible effect of coupling is expected.
The bi-fidelity surrogate is constructed using 476 low-fidelity samples at the maximum convective trajectory point. A set of ten high-fidelity samples are then evaluated at skeleton points identified by the QR factorization, with seven samples used in the BF-ID construction and three samples used to evaluate held-out errors, revealing a maximum error of 2.19%. Sampling of the BF-ID surrogate then permits the construction of a global polynomial chaos expansion (PCE) surrogate model. Using the data up to this point, a 1% failure boundary is identified at a bondline temperature of 580 K. To improve the surrogate accuracy in the region of the failure boundary, a local surrogate was constructed using new HF sample points gathered at parameter values identified by the global surrogate to lie in a “buffer region” near the failure boundary. These new HF samples were then used to construct a local surrogate, part of a global-local hybrid surrogate (GLHS), which uses the global surrogate over most of parameter space and switches to a local surrogate for improved accuracy just in the region that is critical for failure estimation. Using a greedy sampling approach, only 35 HF samples were required for the local surrogate to reach an improved error relative to the global surrogate. In total, only 60 HF samples were required to perform failure estimation and efficiently localize the 1% failure boundary, demonstrating the promise of this technique for failure estimation when model evaluation is computational demanding, as is typically the case for entry system modeling.
Conclusion
Uncertainty quantification efforts relating to the Dragonfly mission’s entry into Titan’s atmosphere were summarized, with particular emphasis on quantifying the reliability of the forebody heat shield. A bi-fidelity interpolative decomposition surrogate model is shown to permit computationally efficient estimation of the failure boundary in parameter space — defined as the point at which the bondline temperature exceeds some critical value. Local refinement near the failure surface using a global-local hybrid surrogate further improves the accuracy with which the failure boundary is localized, significantly reducing the error at the limiting QoI values where failure occurs. This demonstration paves the way for future studies that include higher fidelity modeling of material response and which consider the evolution of heat soak into the TPS across the entire trajectory, rather than just at individual trajectory points.
References
[1] H. Cheng et al. “On the Compression of Low Rank Matrices”. In: SIAM Journal on Scientific Computing 26.4 (2005), pp. 1389–1404. doi: 10.1137/030602678; J. Hampton et al. “Practical error bounds for a non-intrusive bi-fidelity approach to parametric/
stochastic model reduction”. In: Journal of Computational Physics 368 (2018), pp. 315–332. doi:10.1016/j.jcp.2018.04.015
Summary
Sensitivity and uncertainty analyses of the Dragonfly capsule using CFD and radiation solvers, focusing on the inference of atmospheric CH4 concentration and on the estimation of forebody heat shield reliability.