Uncertainty Quantification and Sensitivity Analysis

Computational modeling is an essential tool for designing and understanding complex aerospace systems, from hypersonic vehicles to advanced electric propulsion systems operating in extreme plasma environments. However, the accuracy of these predictions is inherently limited by uncertainties in the physical models used to describe these systems, numerical methods implemented in computational tools, and existing experimental data. Uncertainty Quantification (UQ) provides a framework for identifying, characterizing, and reducing these uncertainties to improve confidence in computational predictions of quantities of interest (QoIs).

UQ is the science of determining how uncertainties in models, numerical algorithms, experiments, and input parameters influence the uncertainty of predicted outcomes. In many aerospace applications, the governing physics are highly complex and involve processes that are not fully understood, such as chemical kinetics, energy exchange mechanisms, radiation physics, and plasma interactions. These limitations introduce uncertainty into the models used within computational tools, which can ultimately impact the accuracy of predicted QoIs, such as heat flux, surface temperature, radiation, and plasma properties.

Sensitivity Analysis (SA) is a complementary technique within UQ that examines how variations in model inputs contribute to changes in a QoI. By identifying which parameters have the greatest influence on model predictions, sensitivity analysis helps determine the dominant sources of uncertainty and guides efforts to improve physical models, experimental measurements, and computational approaches.

Together, UQ and SA enable researchers to quantify the reliability of computational predictions and understand the factors that limit their accuracy. In hypersonics and electric propulsion applications, these tools are particularly valuable because predictions often depend on uncertain physics models describing nonequilibrium gas and plasma dynamics. By propagating uncertainties through high-fidelity computational models and identifying the parameters that most strongly influence predictions of QoIs, UQ and SA provide insight into model limitations and help prioritize future research efforts to improve predictive capabilities.

Air-Breathing Hall Thrusters

Air-Breathing Hall Thrusters

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

Asa O’Neal

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.