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Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. In this work, we defined conditions for discovering new invariant subspaces in the state space of a dynamical system starting from an invariant subspace based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subspaces in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subspaces of the dynamical system and time-series data from any one of the invariant subspaces. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subspaces can be combined to approximate the global Koopman operator.

42 ENGINEERING↗

STITCHES: creating new scenarios of climate model output by stitching together pieces of existing simulations

Abstract. Climate model output emulation has long been attempted to support impact research, mainly to fill in gaps in the scenario space. Given the computational cost of running coupled earth system models (ESMs), which are usually the domain of supercomputers and require on the order of days to weeks to complete a century-long simulation, only a handful of different scenarios are usually chosen to externally force ESM simulations. An effective emulator, able to run on standard computers in times of the order of minutes rather than days could therefore be used to derive climate information under scenarios that were not run by ESMs. Lately, the necessity of accounting for internal variability has also made the availability of initial-condition ensembles, under a specific scenario, important, further increasing the computational demand. At least so far, emulators have been limited to simplified ESM-like output, either seasonal, annual, or decadal averages of basic quantities, like temperature and precipitation, often emulated independently of one another. With this work, we propose a more comprehensive solution to ESM output emulation. Our emulator, STITCHES, uses existing archives of earth system models' (ESMs) scenario experiments to construct ESM-like output under new scenarios or enrich existing initial-condition ensembles, which is what other emulators also aim to do. Importantly, however, STITCHES' output has the same characteristics of the ESM output it sets out to emulate: multivariate, spatially resolved, and high frequency, representing both the forced component and the internal variability around it. STITCHES extends the idea of time sampling – according to which climate outcomes are stratified by the global warming level at which they manifest themselves, irrespective of the scenario and time at which they occur – to the construction of a continuous history of ESM-like output over the whole 21st century, consistent with a 21st-century trajectory of global surface air temperature (GSAT) derived from the scenario that has been chosen as the target of the emulation. STITCHES does so by first splitting the target GSAT trajectory into decade-long windows, then matching each window in turn to a decade-long window within an existing model simulation from the available scenario runs according to its proximity to the target in absolute size of the temperature anomaly and its rate of change. A look-up table is therefore created of a sequence of existing experiment–time-window combinations that, when stitched together, create a GSAT trajectory “similar” to the target. Importantly, we can then stitch together much more than GSAT from these windows, i.e., any output that the ESM has saved for these existing experiment–time-window combinations, at any frequency and spatial scale available in its archive. We show that the stitching does not introduce artifacts in the great majority of cases (we look at temperature and precipitation at monthly frequency and on the native grid of the ESM and at an index of ENSO activity, the Southern Oscillation Index). This is true even if the criteria for the identification of the decades to be stitched together are chosen to work for a smoothed time series of annual GSAT, a result we expect given the larger amount of noise affecting most other variables at finer spatial scales and higher frequencies, which therefore are more “forgiving” of the stitching. We successfully test the method's performance over many ESMs and scenarios. Only a few exceptions surface, but these less-than-optimal outcomes are always associated with a scarcity of the archived simulations from which we can gather the decade-long windows that form the building blocks of the emulated time series. In the great majority of cases, STITCHES' performance is satisfactory according to metrics that reward consistency in trends, interannual and inter-ensemble variance, and autocorrelation structure of the time series stitched together. The method therefore can be used to create ESM-like output according to new scenarios, on the basis of a trajectory of GSAT produced according to that scenario, which could be easily obtained by a simple climate model. It can also be used to increase the size of existing initial-condition ensembles. There are aspects of our emulator that will immediately disqualify it for specific applications, like when climate information is needed whose characteristics result from accumulated quantities over windows of times longer than those used as pieces by STITCHES, droughts longer than a decade for example. But for many applications, we argue that a stitched product can satisfy the climate information needs of impact researchers. STITCHES cannot emulate ESM output from scenarios that result in GSAT trajectories outside of the envelope available in the archive, nor can it emulate trajectories with shapes different from existing ones (overshoots with negative derivative, for example). Therefore, the size and characteristics of the available archives of ESM output are the principal limitations for STITCHES' deployment. Thus, we argue for the possibility of designing scenario experiments within, for example, the next phase of the Coupled Model Intercomparison Project according to new principles, relieved of the need to produce a number of similar trajectories that vary only in radiative forcing strength but more strategically covering the space of temperature anomalies and rates of change.

54 ENVIRONMENTAL SCIENCES↗

TPSAS-NF1676L-13678-DND

An area that shows promise in enhancing structural integrity of aircraft and aerospace structures is integral stitched composite technology. The most recent generation of this technology is the Pultruded Rod Stitched Efficient Unitized Structure (PRSEUS) concept. The goal of the PRSEUS concept is to provide damage-containment capability for composite structures while reducing overall structural weight. The National Aeronautics and Space Administration, the Boeing Company, and the Federal Aviation Administration (FAA) have partnered in an effort to assess the damage-containment features of a full-scale curved PRSEUS panel using the FAA Full-Scale Aircraft Structural Test Evaluation and Research facility. The background, test plan, fixture modifications, pre-test analysis, and planned experimental procedure were presented at the 2011 Aircraft Airworthiness and Sustainment Conference. This follow-on paper and presentation will provide details of the experimental procedure, test results, nondestructive inspection results, and preliminary test analysis comparison. The test program included three phases of loading and inspections: Phase I, as-built; Phase II, with barely visible impact damage (BVID); and Phase III, with a two-bay notch severing the central stiffener. Axial tension, internal pressure, and combined axial tension and internal pressure load conditions were applied during each phase using load levels that demonstrate compliance with the strength, deformation, and damage-tolerance requirements of Title 14 Code of Federal Regulations Part 25. Pressure loads were based on an operating pressure of 9.2 psi, designated as 1P, and the axial loads were based on a design limit load (DLL) of 227 kip. The Phase I test results provided a baseline. First, 50 percent limit-load levels were applied for each of the three load cases to verify proper load introduction and repeatability by examining strain and displacement results. Next, three limit-load cases were applied: limit pressure of 12.2 psi (1.33 P), axial DLL of 227 kip, and combined 1P pressure and axial DLL. Linear strain and displacement results were observed in all load cases. For Phase-II testing, the panel was impacted to introduce BVID with an energy of 40 ft-lb, with the impact located between the central stringer stitch line and flange edge. The impact created non-penetrating visual damage with fiber breaks and delamination between the skin and stringer flange. Limit-load conditions were applied followed by pressure overload of 18.4 psi (2P) and ultimate load conditions, defined as 1.5 times DLL. Inspections were performed after each load case; no damage growth was observed. For Phase-III testing, a two-bay notch was machined severing the central stiffener. The panel was then subjected to limit-load conditions, followed by combined 1P pressure while increasing axial tension load to catastrophic failure. As limit combined loads were approached, damage initiation was observed in the form of 45? splitting at 1P + 79% DLL. Axial loading was increased to 1P + 150 percent DLL causing damage to propagate to the adjacent stringer flanges. Damage was contained within the two-bay region by the stitching rows up to 1P + 176% DLL. Axial load was further increased until catastrophic failure, when a stringer rod failed at 1P + 192% DLL. These test results further verify the damage containment features of the PRSEUS concept and suggest its appropriateness for future flight vehicles.

Andrew Bergan↗

Full-Scale Test and Analysis of a PRSEUS Fuselage Panel to Assess Damage-Containment Features

Stitched composite technology has the potential to substantially decrease structural weight through enhanced damage containment capabilities. The most recent generation of stitched composite technology, the Pultruded Rod Stitched Efficient Unitized Structure (PRSEUS) concept, has been shown to successfully arrest damage at the sub-component level through tension testing of a three stringer panel with damage in the form of a two-bay notch. In a joint effort undertaken by the National Aeronautics and Space Administration (NASA), the Federal Aviation Administration (FAA), and the Boeing Company, further studies are being conducted to characterize the damage containment features of the PRSEUS concept. A full-scale residual strength test will be performed on a fuselage panel to determine if the load capacity will meet strength, deformation, and damage tolerance requirements. A curved panel was designed, fabricated, and prepared for residual strength testing. A pre-test Finite Element Model (FEM) was developed using design allowables from previous test programs to predict test panel deformation characteristics and margins of safety. Three phases of testing with increasing damage severity include: (1) as manufactured; (2) barely visible impact damage (BVID) and visible impact damage (VID); and (3) discrete source damage (DSD) where the panel will be loaded to catastrophic failure. This paper presents the background information, test plan, and experimental procedure. This paper is the first of several future articles reporting the test preparations, results, and analysis conducted in the test program.

Bergan, Andrew↗

Trajectory Engineering with Modular Patched Conics for Entry Systems and TPS (TEMPEST)

Brief Presenter Biography (35 word limit): Bohdan Wesely is an Aerospace Engineer in the Entry Systems and Technology Division at Ames. He has worked on a variety of projects for NASA including integrated TPS (thermal protection system) flight hardware deliveries and testing services for commercial partners. Introduction: TEMPEST is a new trajectory analysis framework that is designed to fill the gap between dedicated flight mechanics tools and aerothermal and TPS sizing tools. The project started as an SJSU master’s thesis and has since evolved into a general conceptual design tool capable of studying a wide variety of entry problems. Development is ongoing in the Entry Systems and Technology Division at NASA ARC. Why TEMPEST: Space missions involving entry into a planetary atmosphere involve a series of unique requirements across multiple disciplines. Whether it is traditional entry descent and landing (EDL), or aerocapture, the vehicle must navigate to its target landing location or orbit state, and the TPS must protect the payload during entry. The design process typically involves iterative handoffs between various flight mechanics, flow solver, and material response level tools. During the early conceptual phase, a wide variety of feasible trajectories are simulated in a Monte Carlo scenario which broadly satisfy the mission or landing requirements. Next, computational fluid dynamics (CFD), direct simulation Monte Carlo (DSMC), and other flow solver analyses are performed at various key trajectory points to generate an aero-database, heating and TPS design requirements also emerge at this stage. At this point, with updated aerodynamics from the various flow solvers, trajectories can be re-run, this in turn can change the required freestream conditions for the CFD tools, and as a project progresses, these analyses converge, and uncertainty is reduced. However, there is always a “hand-off” occurring between two inherently coupled phenomena. Analysis Description: One of the goals with TEMPEST is to use a variety of first principles estimation methods coupled with an atmosphere model to predict vehicle aerothermodynamics across the entire flight regime while propagating a 3 or 6 degree of freedom (DoF) trajectory. Aerodynamics methods include modified Newtonian, Maxwell and Cercignani- Lampis-Lord (CLL) for continuum, transitional, and free molecular flow regimes. Aerothermodynamics include boundary layer and reference enthalpy methods, and Mutation++ for non-equilibrium chemistry modeling. TEMPEST is also capable of stitching multiple trajectory segments together to study mission scenarios like multi-pass aerocapture and aero-gravity assists. Most of the program is implemented in MATLAB using modern system objects, it relies on several C++ shared libraries for supporting tools like Gmsh, the Global Reference Atmospheric Model (GRAM), and Mutation++. The various first principles aerothermal estimation methods are discretized across either a structured axisymmetric panel mesh or an unstructured tri-mesh generated from an open-source tool such as Gmsh, this allows solutions on the same mesh to be compared across tools such as CB-Aero. CFD Coupling. A physics-aware, gaussian process CFD anchoring scheme is proposed to adjust the various first principles methods as a CFD database is populated. One goal for this anchoring module is to inform the project where CFD should be run. Full knowledge of the entire trajectory, atmosphere, and aerothermodynamics allows for easier identification of high sensitivity areas and uncertainty quantification. While the first principles effects are well known and proven accurate in existing tools such as CB- Aero and Cart3D, a physics aware CFD anchoring scheme increases tool credibility across a project lifecycle. Material Response Modeling. Correct TPS sizing is critical for optimizing mass for science payloads and ensuring mission success. The process typically involves a thermal analysis along the trajectory with surface heating environments as a boundary condition. Several design constraints are maximum bondline temperature and maximum recession with various margining techniques. The material response tool FIAT, developed out of NASA Ames, is currently being integrated into the TEMPEST environment. TPS recession, shape change, mass loss, and mass property alteration are all factors that can perturb an entry trajectory. For missions like Mars 2020, recession was minimal and was safely handled separately as a post process. For missions such as Jupiter Galileo with a high TPS mass fraction or asteroid entries, recession plays a major role. The proposed fully coupled scheme is to use an epoch-based approach where the trajectory integration is halted after a recession threshold, the energy balance and FIAT are solved at each panel, the mesh, aerodynamics, and mass properties are updated, and the trajectory continues. Several computational tradeoffs have been made during the development of TEMPEST to limit the cost of a single trajectory and preserve its utility as a conceptual, rapid iteration tool. Conclusion: Development of TEMPEST is ongoing and the project is still in its infancy. This talk aims to showcase its unique capabilities to support future NASA entry systems missions.

Bohdan O Wesely↗