Uncertainty Quantification and Propagation

Introduction

In numerical modeling and simulation, some degree of uncertainty is inevitable in the ability of the model to truly describe the physics of interest and/or in the data this model uses to assist in describing these physical phenomena For this reason, and because numerical predictions are often the basis of engineering decisions, uncertainty quantification has been a subject of concern for many years. With the advent of multiscale and multi-physics modeling, finding practical and yet rigorous ways to interpret uncertainty and characterize its impact on the assessment of probable outcomes has become ever more challenging. Numerous techniques have been developed based on stochastic modeling of uncertainties and their quantification in computational models. An introduction to uncertainties and a general description of such techniques relevant to re-entry analysis and is discussed in this section.

In general, two different types of uncertainty exist,

1. Aleatory Uncertainty
   • These kind of uncertainties concern physical phenomena which are random by nature, like the experimental measurements of pressure and temperature of a flow-field [1].
   • Also called as statistical uncertainty.
2. Epistemic Uncertainty
   • These kind of uncertainties concern the parameters of a computational model, for which there is a lack of knowledge, and also the modeling errors [1]. Like the lack of knowledge in a physical phenomena.
   • Also called as systemic uncertainty.

The techniques from probability theory and mathematical statistics of uncertainty quantification do not distinguish these different types of uncertainties as the tools available are applicable to either types [1].

Uncertainties during re-entry and risk analysis

Object oriented re-entry analysis (low-fidelity) tools are primarily used to calculate risk due to various factors during re-entry of a spacecraft or an object. various models that are used to describe the re-entry physics use simplifying assumptions that reduce the reliability of their predictions. The subsequent mismodeling errors induce uncertainties in the risk metrics (number of fragments, on-ground risk or casualty area). Moreover, the lack of knowledge of the reentry scenario (Initial conditions at re-entry interface, the material composition, ballistic coefficient or the mass) also affects the prediction reliability. There are two main sources of uncertainties in a re-entry analysis,

• Uncertainties due to model errors
• Uncertainties on operating conditions and model parameters

Uncertainties due to model errors

( This section will be updated in more detail during the next update to the Stardust-R Wiki )

Different models are used to quantify the physical phenomena based on the level of complexity required. Very conservative modeling is used in object-oriented tools in favor of simplicity and computational time. The following describe various sources of uncertainty due to improper modeling in re-entry,

Aerodynamic and aero-thermal models

• Most re-entry solvers described in this work share similarities in their aerodynamic and aerothermal models. These models are pretty accurate for simple shapes such as spheres, cylinders and flat-plates (primitive shapes) when compared with high-fidelity CFD simulations but, suffer heavily in accuracy for other shapes such as cones.
• Heat flux calculations are usually more challenging to evaluate because of various chemical reactions during re-entry. For eg., for spheres in the hypersonic continuum regime, the agreement is typically good (with experiments) but large errors are observed for flat plates (between 50 and 100 %) due to the mismodeling of the side faces of the plate.
• Errors are majorly due to the bridging functions in rarefied flow regime, sharp edged objects (Invalidity of modified-Newton method), treatment of hidden surfaces in flow-field, Concave surfaces (Improper applicability of modified-Newton method), gas-surface interactions (shock-shock as well), and subsonic flow recirculation zones.

Material modeling and surface effects

• Reactive hypersonic boundary layers are very poorly modeled in re-entry tools. Various chemical reactions which occur at the surface like oxidation reactions, catalytic recombination reactions (significant contribution to convective heat flux) [2][3], ablation process, etc.,.
• Material modeling of composite materials.

Spacecraft models

• To enable simplicity and reduce computational time, a spacecraft is modeled as a combination of primitive shapes. While the error in mass of the object can be made negligible, the moment of inertia matrix and center of mass locations may not be satisfactory.
• Details of sub-system components cannot be modeled accurately.
• These approximations modify the spacecraft’s attitude motion, trajectory and heat-flux.

Fragmentation modeling

• Break-up during re-entry is extremely difficult to model even using high-fidelity structural tools like Finite Element Analysis (FEA) and peridynamics (PD). The existing models assume certain break-up criteria to dissolute object model into its primitive shapes.
• As a consequence, the shape of the fragments and the altitudes at which they are released is subject to uncertainties.

Uncertainties on operating conditions and model parameters

( This section will be updated in more detail during the next update to the Stardust-R Wiki )

This source of uncertainty is independent of model errors and, arises due to the errors in knowledge of model parameters and operating conditions.

Local atmospheric variations

• The atmosphere at various altitudes is governed by a multitude of phenomena. The upper atmosphere is determined by solar and geo-magnetic activity (space weather). The lower atmosphere is influenced by turbulence and winds.
• In the most advanced atmosphere model used in space object reentry simulators (NRLMSISE00, US standard atmosphere 1976), the average atmosphere characteristics (density pressure temperature) are computed for a given time, position, solar activity, and Earth geomagnetic activity that may not be known for a given reentry scenario.

Object position and attitude motions

• The exact attitude motion and position of a reentering space object (at re-entry interface) is not entirely known especially for uncontrolled re-entries. Even the time (epoch) of re-entry is predicted with an uncertainty.
• For controlled re-entries, the uncertainties are usually lower (depends on accuracy of de-orbiting maneuver).
• The object position and ballistic coefficient determination is also based on noisy on-ground observations.

Material characteristics

• Material behavior is measured experimentally, and therefore the values are naturally subject to uncertainty. The properties like thermal conductivity, fusion temperature and emissivity have standard procedures for measurement (their uncertainty is lower). But, the wall catalytic effects [3] and composites ablation rate are complicated.
• Material structural properties like transition to plasticity and buckling limits are not readily known at high temperatures.

Composition of the object/spacecraft

• Exact material composition, mass and geometry of an object is unknown at the time of re-entry. This is more influential for space debris re-entries.
• In-orbit collisions and break-up may modify the geometry of the object and ultimately affects the ballistic coefficient and center of mass of the re-entering object.

Techniques for Modeling uncertainties

1. HDMR-cased sensitivity analysis and uncertainty quantification.
2. A probability density-based approach to the propagation of re-entry uncertainties.
3. Kriging and Self Organizing Maps based adaptive sampling.

References

[1] Soize, Christian. Uncertainty quantification. Springer International Publishing AG, 2017.
[2] Scott, C., and S. Derry. “Catalytic recombination and Space Shuttle heating.” 3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference., 1982.
[3] Yang, Yosheph, Sai Abhishek Peddakotla, Rakesh Kumar, and Gisu Park. “Effect of argon gas in oxygen catalytic recombination on a silica surface: A reactive molecular dynamics study.” Acta Astronautica (2020).