Uncertainty Quantification & Sensitivity Analysis
Introduction
Each decade, NASA releases Decadal Surveys that provide detailed insights into the strategic priorities and mission directives for the next decade across key scientific domains. The 2023-2032 Planetary Science and Astrobiology Decadal Survey has stated that a mission to the Ice Giant planets of Uranus and Neptune is a high priority. These planets are considered the least explored in our solar system, and we have never entered a manmade probe into these atmospheres. In order to design a successful mission to the Ice Giant planets, we must be able to accurately predict quantities of interest (QoIs) for heat shield design, such as the convective and radiative heating the vehicle will experience upon its hypersonic entry into the Ice Giant atmospheres, which are very different from Earth’s atmosphere. It is known that these atmospheres are made up mostly of hydrogen and helium with trace amounts of methane, but the exact chemical composition is uncertain since we have never sent a manmade object into these atmospheres before. These uncertainties have been shown to cause large variations in predictions of the radiative heating that the entry vehicle will experience.
We can simulate the nonequilibrium flow that is characteristic of a planetary entry using tools like Computational Fluid Dynamics (CFD) or an incident shock solver. Experimentally, these flows can be simulated in shock tube facilities, expansion tunnel facilities, and sometimes plasma wind tunnel facilities. Experimental data is useful for validating computational tools that generate simulations. However, in the application of an Ice Giant planetary entry, there are major uncertainties both within the experimental data as well as the chemical kinetics models that are implemented in computational tools. UQ & SA can be helpful in quantifying these uncertainties and understanding the limitations that come with both the experimental data that is generated and the computational tools that are used to simulate these flows.
Research at NGPDL
In NGPDL, mixed UQ is being used to quantify uncertainties associated with both experimental data as well as the chemical kinetics model for hydrogen/helium/methane flows that are characteristic of an Ice Giant entry flow. The mixed UQ problem refers to two different types of uncertainty: aleatory and epistemic. Aleatory uncertainty is due to the inherent randomness of a system, like freestream measurements generated in experiments, and is generally considered to be irreducible. Epistemic uncertainty is generally due to a lack of knowledge in the system. For example, epistemic uncertainty can be represented by the lack of knowledge in the hydrogen/helium/methane chemical kinetics model that is implemented in the numerical tool that is used to simulate these flows. The mixed UQ problem is set up in a nested loop such that the aleatory uncertain parameters are varied in the inner loop and the epistemic uncertain parameters are varied in the outer loop.
The aleatory and epistemic uncertainties are propagated through the model to compute a surrogate model through a polynomial chaos expansion (PCE). From this, Sobol’ indices can be computed, which essentially quantify how much an uncertain input parameter contributes to variation in a QoI. The QoI in the Ice Giant entry problem is spectral radiance, since emission spectra is what is experimentally generated. Figure 1 shows an example of Sobol' indices being computed for the integrated spectral radiance that was experimentally measured in the T6 Stalker Tunnel at the University of Oxford. This particular problem has 22 uncertain parameters, but it can be observed on the plot that only a few of them have non-negligible Sobol' indices, indicating that these are the only parameters that contribute to variation in the spectral radiance (which is the QoI). This is helpful for knowing which parameters need to be tuned to improve chemical kinetics models for future work.
Figure 1. Sobol' indices for uncertain epistemic chemical kinetic parameters and aleatory freestream parameters for an Ice Giant entry flow.
Investigators
Acknowledgements
This work is supported by the National Science Foundation Graduate Research Fellowship Program (NSF GRFP).
References
[1] Leszcz, C., Aiken, T.T., and Boyd, I.D., “Mixed Uncertainty Quantification for Hypersonic Entry Into Hydrogen-Helium Atmospheres,” AIAA AVIATION 2026 Forum, AIAA Paper , July 2026.
Introduction
Hall effect thrusters (HETs) are a widely used electric propulsion (EP) technology for in-space applications. HETs have been developed and operated for decades and achieve high thrust efficiency when using traditional propellants, primarily xenon. Now, there is an interest in developing HETs that operate on alternative propellants. There is a growing interest in using Air-Breathing Electric Propulsion (ABEP) in very low Earth orbit (VLEO) to compensate for the significant drag experienced by spacecraft due to the relatively high atmospheric density. Operation of a HET on air propellants (nitrogen and oxygen) poses several significant challenges that the EP community is working to address. Many of these challenges are associated with the molecular nature of air, differing from xenon which exists naturally in atomic form and is chemically inert. When operating a HET on molecular propellants, inelastic processes (such as rovibrational excitation and dissociation) combined with lower ionization cross sections and higher ionization energies significantly reduce thruster performance. Furthermore, since most HET development has been performed using xenon and other noble gas propellants, there is very little experimental data for HETs operating on air species. If ABEP is to become a viable technology for long-duration VLEO missions, it is essential that molecular-specific inefficiencies be characterized and mitigated.
Modeling will play an essential role in the development and qualification of ABEP systems for a few reasons. First, it is impossible to fully recreate the VLEO environment using ground-based test facilities, meaning models will be required to supplement experiments. Second, models can provide useful diagnostics that would be difficult to measure experimentally. Finally, parametric modeling allows for inexpensive and rapid evaluation of large parameter spaces and can inform scaling laws for ABEP system design.
Despite the advantages offered by computational models, we lack the ability to fully predict HET performance ab-initio for several reasons. First, the physics governing cross-field electron transport is still not completely understood. This is commonly referred to as the anomalous transport problem and introduces the need for a closure model with free parameters. These free parameters have typically been tuned to match performance metrics such as thrust and discharge current. Furthermore, existing models remain underdeveloped for molecular propellants since most have limited support for molecular chemistry, such as rovibrational excitation, dissociation, etc. Since it is impractical to implement a complete set of chemistry (such as a state-to-state treatment) in an engineering HET model, there is a need to assess the sensitivity to different reaction pathways in the pursuit of a reduced and practical chemistry model.
Sensitivity analysis (SA) and uncertainty quantification (UQ) methods provide powerful tools for addressing these modeling deficiencies. Bayesian inference can be used to infer the free anomalous transport parameters with corresponding uncertainty. SA provides an opportunity to quantify the relative impacts of different reaction pathways on model predictions. These methods together will be invaluable in the pursuit of effective computational tools for ABEP system development.
Bayesian Inference
HET models rely on closure models for the still unresolved anomalous electron transport. This introduces free parameters that must be calibrated and often do not generalize well to other thrusters and operating conditions. For atomic propellants (such as xenon), the transport parameters have typically been tuned using optimization methods, where an objective function is defined to minimize error against observables like thrust and discharge current. For molecular species, additional degrees of freedom are introduced that make it more difficult to identify the anomalous transport parameters. Bayesian inference offers an intelligent way to solve the inverse problem and infer probability distributions of these free parameters from macroscopic observables.
Our work has focused on the P5 HET operating on nitrogen, which is subject to dissociation and molecular-specific excitation modes. Since N2 can dissociate, the species involved are neutral and ionized N2 and N. This work has incorporated a reduced nitrogen chemistry model to the open-source code HallThruster.jl and uses species-resolved efficiencies as calibration observables. Preliminary results suggest that the introduction of species-resolved efficiencies make the transport parameters identifiable under a Bayesian calibration, enabling predictions of quantities of interest with corresponding uncertainty.
Sensitivity Analysis
SA quantifies how uncertainty in a model's outputs depends on uncertainty in its inputs. A common measure of sensitivity is the Sobol index, which quantifies how much of a model’s output variance is attributed to specific inputs and the interactions between inputs. The total order Sobol index quantifies the contribution of a specific input and all its interaction effects. In the context of developing a HET model that incorporates air propellants, it is useful to understand the sensitivity of the model to different reaction pathways. By identifying the inputs that contribute most to output uncertainty, calibration efforts can be focused where they will have the greatest impact. Furthermore, SA may identify inputs with negligible influence on the quantities of interest, enabling dimensionality reduction and more efficient computation. Finally, for experimentally derived inputs such as cross sections used for computing reaction rates, SA can highlight where experimentalists should focus their time, since reducing the uncertainty in the most sensitive model inputs leads the largest reduction in model output uncertainty.
This work has focused on assessing the sensitivity of several quantities of interest (QoIs) to the anomalous transport parameters and chemical rate coefficients. The nitrogen chemistry model incorporates 46 electron-neutral reaction pathways, including excitations (rotational, vibrational, and electronic), dissociation, elastic collisions, and ionization reactions. Additionally, sensitivity to the anomalous transport parameters is investigated since these parameters are a known source of uncertainty. This brings the total number of uncertain parameters to 50. Preliminary results indicate that the investigated QoIs are most sensitive to the ionization rates, dissociation rates, and anomalous transport parameters.
Investigators
Acknowledgements
This work is supported by the National Science Foundation Graduate Research Fellowship Program (NSF GRFP). This work utilizes the open-source code HallThruster.jl, which is developed and maintained by researchers at the University of Michigan.