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At least 577 records · Page 32

A unified hp-HDG framework for Friedrichs' PDE systems

Here, this work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

97 MATHEMATICS AND COMPUTING↗

Monte Carlo analysis of the Titan III/Transfer Orbit Stage guidance system for the Mars Observer mission

An important part of space launch vehicle mission planning for a planetary mission is the integrated analysis of guidance and performance dispersions for both booster and upper stage vehicles. For the Mars Observer mission, an integrated trajectory analysis was used to maximize the scientific payload and to minimize injection errors by optimizing the energy management of both vehicles. This was accomplished by designing the Titan III booster vehicle to inject into a hyperbolic departure plane, and the Transfer Orbit Stage (TOS) to correct any booster dispersions. An integrated Monte Carlo analysis of the performance and guidance dispersions of both vehicles provided sensitivities, an evaluation of their guidance schemes and an injection error covariance matrix. The polynomial guidance schemes used for the Titan III variable flight azimuth computations and the TOS solid rocket motor ignition time and burn direction derivations accounted for a wide variation of launch times, performance dispersions, and target conditions. The Mars Observer spacecraft was launched on 25 September 1992 on the Titan III/TOS vehicle. The post flight analysis indicated that a near perfect park orbit injection was achieved, followed by a trans-Mars injection with less than 2sigma errors.

Bell, Stephen C.↗

Integrability, normal forms, and magnetic axis coordinates

Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation entirely consisting of invariant submanifolds. A compact regular leaf (a flux surface) of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves, including elliptic and hyperbolic magnetic axes. In this work, the existence of exact smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydrodynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields: they are presymplectic systems.

97 MATHEMATICS AND COMPUTING↗

Universal Lambert and Kepler algorithms for autonomous rendezvous

This paper describes Lambert and Kepler algorithms designed to be the core of an autonomous rendezvous guidance system for an onboard computer. Applications include robotic and piloted missions to the moon and planets. Flight software must be compact, fast, and totally reliable. Although high accuracy is not essential for flight, in double precision these algorithms are accurate to at least 14 places almost everywhere. Both are universal; they apply to elliptic, parabolic, hyperbolic, and even rectilinear trajectories. The algorithms are improvements to those published by Battin (1987).

Klumpp, Allan R.↗

OSIRIS-REx Encounters Bennu: Initial Assessment from the Approach Phase

The OSIRIS-REx spacecraft launched on September 8, 2016, on a seven-year journey to return samples from asteroid (101955) Bennu. This presentation summarizes the scientific results from the Approach and Preliminary Survey phases. Bennu observations are set to begin on August 17, 2018,when the asteroid is bright enough for detection by the PolyCam. PolyCam and MapCam collect data to survey the asteroid environment for any hazards and characterize the asteroid point-source photometric properties. Resolved images acquired during final approach, starting in late October 2018, allow the creation of a shape model using stereophotoclinometry (SPC), needed by both the navigation team and science planners. The OVIRS and OTES spectrometers characterize the point- source spectral properties over a full rotation period, providing a first look at any features and thermophysical properties. TAGSAM is released from the launch container and deployed into the sampling configuration then returned to the stow position.Preliminary Survey follows the Approach Phase in early December 2018. This phase consists of a series of hyperbolic trajectories that cross over the North and South poles and the equator of Bennu at a close-approach distance of 7 km. Images from these Preliminary Survey passes provide data to complete the 75-cm resolution SPC global shape model and solve for the rotation state. Once the shape model is complete, the asteroid coordinate system is defined for co-registration of all data products. These higher-resolution images also constrain the photometric properties and allow for an initial assessment of the geology. In Preliminary Survey the team also obtains the first OLA data, providing a measure of the surface topography. OVIRS and OTES collect data as "ride-along" instruments, with the spacecraft pointing driven by imaging constraints. These data provide a first look at the spectral variation across the surface of Bennu. Radio science measurements, combined with altimetry and imagery, determine Bennu's mass, a prerequisite to placing the spacecraft into orbit in late December 2018. Together, data from the Approach and Preliminary Survey phases set the stage for the extensive mapping planned for 2019. These dates are the baseline plan. Any contingency or unexpected discovery may change this mission profile.

Lauretta, Dante S.↗

An automatic multigrid method for the solution of sparse linear systems

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

Shapira, Yair↗

Morphology Control of Self-Assembled Three-Phase Au-BaTiO 3 –ZnO Hybrid Metamaterial for Tunable Optical Properties

Microstructural control in metal-dielectric hybrid metamaterials presents enormous opportunities in tailoring the physical properties including the magnetic and optical properties. In this paper, we demonstrate a strong tunability achieved in the microstructure of self-assembled ordered three-phase Au-BaTiO 3 –ZnO hybrid metamaterial along with its optical properties, grown by a pulsed laser deposition method. Varying the growth temperature, deposition frequency, and template thickness evolves the microstructure by tuning the Au and ZnO pillar geometry as well as the shape and size of the Au nanoparticles capping the ZnO nanowires. The three-phase hybrid metamaterials exhibit unique optical properties, including enhanced nonlinear optical properties, hyperbolic dispersion in the visible and near-infrared wavelength region, and tuned epsilon-near-zero (ENZ) wavelength upon varying the deposition parameters. This study suggests that the three-phase hybrid metamaterials present great potential in the microstructure and optical property tuning that can also be applied to other two-phase and three-phase nanocomposite systems.

36 MATERIALS SCIENCE↗

Physical and dynamical characterization of hyperbolic comet C/2017 U7 (PANSTARRS)

Here we present here a dynamical and observational study of the comet C/2017 U7 (PANSTARRS). This comet was discovered in 2017 and found to have a hyperbolic orbit. Our dynamical analysis shows that the object has probably originated in the Oort cloud, however an interstellar origin cannot be discarded. The observations were obtained in 2018 and 2019 using the Goodman High Throughput Spectrograph (GHTS) at the SOAR telescope. We obtained visible spectra covering the wavelength range of 0.5 - 0.9 μm and also images in the SDSS filters system. Both the low-resolution reflectance spectrum and the reflectance spectra derived from the SDSS filters show an atypical band at ~ 0.595 μm. We conducted a comparative study of the colors and reflectance spectra of different small body populations (e.g., comets, Centaurs, and trans-Neptunian objects or TNOs) from the literature and concluded that the spectra and the colors of this comet are atypical, showing only some overlap with those of some known members of the TNOs and Centaurs, within the large uncertainties of the measurements of those populations. It is found that the feature and overall spectral shape can be reproduced by laboratory spectra of kerite, a template for aliphatic-rich hydrocarbons that has been previously identified in NIR cometary spectra absorptions. It is tentatively proposed that the unusual spectral shape is the result of a particle size distribution of dust grains in the coma or on the surface that has arisen due to a low grain ejection velocity from the surface and large nucleus size.

79 ASTRONOMY AND ASTROPHYSICS↗

Evaluation of image quality in a Cassegrain-type telescope with an oscillating secondary mirror

A ray-trace analysis is described of aberrations and extreme rays of a Cassegrain-type telescope with a tilted secondary mirror. The work was motivated by the need to understand the factors limiting image quality and to assist in the design of secondary mirrors for three telescopes with oscillating secondary mirrors (OSM) used at Ames Research Center for high altitude infrared astronomy. The telescopes are a 31-cm-diameter Dall-Kirkham (elliptical primary, spherical secondary) flown aboard a Lear jet, a 71-cm balloon-borne Dall-Kirkham flown on the AIROscope gondola, and a 91-cm true Cassegrain (parabolic primary, hyperbolic secondary) flown aboard a C-141 jet transport. The optics for these telescopes were not designed specifically for OSM operation, but all have OSM's and all must be used with various detector configurations; therefore, a facility that evaluates the performance of a telescope for a given configuration is useful. The analytical expressions are summarized and results for the above systems are discussed. Details of the calculation and a discussion of the computer program are given in the appendices.

Erickson, E. F.↗

Far field boundary conditions for computation over long time

A new class of computational far field boundary conditions for hyperbolic partial differential equations is developed. These boundary conditions combine properties of absorbing boundary conditions for transient solutions and properties of far field boundary conditions for steady-state problems. The conditions can be used to limit the computational domain when both traveling waves and evanescent waves are present. Boundary conditions for scalar wave equations are derived and analyzed. Extensions to systems of equations are discussed and results from numerical experiments are presented.

Engquist, Bjorn↗

Variable Step Integration Coupled with the Method of Characteristics Solution for Water-Hammer Analysis, A Case Study

One-dimensional water-hammer modeling involves the solution of two coupled non-linear hyperbolic partial differential equations (PDEs). These equations result from applying the principles of conservation of mass and momentum to flow through a pipe, and usually the assumption that the speed at which pressure waves propagate through the pipe is constant. In order to solve these equations for the interested quantities (i.e. pressures and flow rates), they must first be converted to a system of ordinary differential equations (ODEs) by either approximating the spatial derivative terms with numerical techniques or using the Method of Characteristics (MOC). The MOC approach is ideal in that no numerical approximation errors are introduced in converting the original system of PDEs into an equivalent system of ODEs. Unfortunately this resulting system of ODEs is bound by a time step constraint so that when integrating the equations the solution can only be obtained at fixed time intervals. If the fluid system to be modeled also contains dynamic components (i.e. components that are best modeled by a system of ODEs), it may be necessary to take extremely small time steps during certain points of the model simulation in order to achieve stability and/or accuracy in the solution. Coupled together, the fixed time step constraint invoked by the MOC, and the occasional need for extremely small time steps in order to obtain stability and/or accuracy, can greatly increase simulation run times. As one solution to this problem, a method for combining variable step integration (VSI) algorithms with the MOC was developed for modeling water-hammer in systems with highly dynamic components. A case study is presented in which reverse flow through a dual-flapper check valve introduces a water-hammer event. The predicted pressure responses upstream of the check-valve are compared with test data.

Turpin, Jason B.↗

A-Posteriori Error Estimation for Hyperbolic Conservation Laws with Constraint

This lecture considers a-posteriori error estimates for the numerical solution of conservation laws with time invariant constraints such as those arising in magnetohydrodynamics (MHD) and gravitational physics. Using standard duality arguments, a-posteriori error estimates for the discontinuous Galerkin finite element method are then presented for MHD with solenoidal constraint. From these estimates, a procedure for adaptive discretization is outlined. A taxonomy of Green's functions for the linearized MHD operator is given which characterizes the domain of dependence for pointwise errors. The extension to other constrained systems such as the Einstein equations of gravitational physics are then considered. Finally, future directions and open problems are discussed.

Barth, Timothy↗

Geometry and properties of the plasma in a quadrupole ion pump

The diameter of the plasma in a quadrupole ionization gauge and quadrupole ion pump has been measured as a function of pressure. The radio-frequency quadrupole systems are tuned in such a way that electrons remain on stable trajectories in the center of the tube. Being emitted at one end, the electrons encounter an inverted field at the other end so that they travel on spiral paths back and forth until they collide with the rest gas molecules, producing ions which are collected as an ion current at a cylinder surrounding the four hyperbolically shaped quadrupole electrodes.

Schwarz, H.↗

A comparative study of the parabolized Navier-Stokes code using various grid-generation techniques

The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic, and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6 deg and 20 deg, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching step sizes with the implicit shock fitting procedure. A correlation is found between the marching step size, Reynolds number, and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.

Kaul, U. K.↗

A comparative study of the parabolized Navier-Stokes code using various grid-generation techniques

The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6 deg and 20 deg, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching-step sizes with the implicit shock fitting procedure. A correlation is found between the marching-step size, Reynolds number and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.

Kaul, U. K.↗

Insights on the continuous representations of piecewise-smooth nonlinear systems: limits of applicability and effectiveness

Dynamical systems subject to intermittent contact are often modeled with piecewise-smooth contact forces. However, the discontinuous nature of the contact can cause inaccuracies in numerical results or failure in numerical solvers. Representing the piecewise contact force with a continuous and smooth function can mitigate these problems, but not all continuous representations may be appropriate for this use. Here, five representations used by previous researchers (polynomial, rational polynomial, hyperbolic tangent, arctangent, and logarithm-arctangent functions) are studied to determine which ones most accurately capture nonlinear behaviors including super- and subharmonic resonances, multiple solutions, and chaos. The test case is a single-DOF forced Duffing oscillator with freeplay nonlinearity, solved using direct time integration. This work intends to expand on past studies by determining the limits of applicability for each representation and what numerical problems may occur.

42 ENGINEERING↗

The evolution of comet orbits

The origin of comets and the evolution of their orbits are discussed. Factors considered include: the law of survival of comets against ejection on hyperbolic orbits; short-period comets are not created by single close encounters of near-parabolic comets with Jupiter; observable long-period comets do not evolve into observable short-period comets; unobservable long-period comets with perihelia near Jupiter can evolve into observable short-period comets; long-period comets cannot have been formed or created within the planetary region of the solar system (excluding the effects of stellar perturbations); it is possible that some of the short-period comets could have been formed inside the orbit of Neptune; circularly-restricted three-body problem, and its associated Jacobi integral, are not valid approximations to use in studying origin and evolution of comets.

Everhart, E.↗

Thermal Protection System to Enable Ice Giant Aerocapture Mission for Delivering Both an Orbiter and an In Situ Probe

The Ice Giants have been identified as high priority science destinations in the last Decadal Survey [1] and could benefit from aerocapture as the primary method for orbit insertion [2]. A mass-efficient aerocapture system will enable the delivery of an orbiter along with an atmospheric probe (for in situ measurements to anchor global data collected by the orbiter) and possibly a lander at Triton [3]. Aerocapture could be executed either using low L/D rigid aeroshell with lift modulation (LMA) [4] or using deployable aeroshell using drag modulation (DMA) [3]. Nearly two decades ago, a NASA-funded team performed Neptune-Triton aerocapture studies with a mid-L/D lifting configuration [5] for achieving orbit using LMA. This study showed aerocapture challenges. Due to very high peak entry conditions combined with very high heat-load, a suite of TPS materials was required and this suite was deemed problematic from a qualification perspective, due test facility limitations. In the past 20 years, progress made in GN&C for lift-guided entry missions such as MSL, Orion EFT1, Mars 2020 and the upcoming Artemis missions, and the expertise in blunt body aerodynamics at large scale (~ 5m) has led the EDL community to conclude that aerocapture is a “go do” engineering activity and most technologies are in hand to propose missions with aerocapture [6] [7]. Aerocapture using DMA, currently in development, is an option for Ice Giant Missions. While DMA is simpler in some sense, due to ballistic entry and no need for lift-guided maneuvering, it has challenges and it’s maturity is lower. LMA and DMA both require one or more ablative Thermal Protection System (TPS) materials for the rigid aeroshell element. The ablative TPS needs to be robust and mass efficient due to the high heat loads and size of the rigid aeroshell. Currently, there are capable ablative thermal protection materials, e.g., Heatshield for Extreme Entry Environments Technology (HEEET), 3-D woven Mid-Density Carbon- Phenolic (3MDCP), and PICA (Phenolic-Impregnated Carbon Ablator) that are mature, i.e., at TRL 6 or higher. NASA also invested in Conformal PICA that was matured to TRL 5. Our goal is to evaluate the applicability of high TRL TPS and consider other design options. We first establish bounding aerocapture trajectories for a wide range of arrival conditions and the associated aerothermal environment. Based on the environments, we then determine the predicted TPS mass for the aeroshell [4]. In this presentation, we will outline the process by which we establish bounding aerocapture trajectories for hyperbolic excess velocities ranging from 27 km/s to 35 km/s, which are shown to be a range of velocities that can reduce the trip time from ~14 years to 8 years. The above velocity range translates to ~12 km/s to ~24 km/s arrival velocities at the planetary entry interface [2]. The velocity reduction required to achieve orbit ranges between ~2.5 km/s to 9.5 km/s for both Neptune and Triton. Propulsive insertion alone, due to the amount of fuel required to achieve the required velocity reduction, limits the science returned [2]. We establish the bounding aerocapture trajectories for a low L/D (~ 0.4) configuration for three different ballistic coefficients. The ballistic coefficient range is determined from three different aeroshell diameters of 3m, 4m and 5m and with an entry system mass of 2200 kg. With the above range of design parameters, we then determine conservative/bounding estimates of aerothermal environments by using a combination of CFD simulations and stagnation point heating estimates [7]. This engineering approach allows us to first assess the TPS need vs. TPS capability and determine the applicability of existing TPS. Once an applicable suite of TPS is determined, the TPS thickness and mass are computed. We show that the TPS mass fraction can be as low as 5% to as high as 20%, depending on total trip time reduction and other design parameters for a range of TPS. This is a large range for TPS mass fraction. We show PICA and HEEET can indeed enable aerocapture missions, but the missions incur a mass penalty. TPS mass savings, can be further reduced with the use of conformal PICA. Advancing the development of Conformal PICA to make it robust across the entire aerothermal environment (peak heat-flux, pressure and shear) range will result in TPS mass fractions of < 10% for Ice Giant aerocapture missions such as the Neptune-Triton mission. Aerocapture allows for not only shortening the trip time but enables larger mass to be placed in orbit. Furthermore, probes deployed from orbit will benefit in reduced entry environments allowing for a lower risk TPS implementation as compared to mission designs where the probe is released prior to orbit insertion. One of the challenges for the Ice Giant community is to ensure mission designs that maximize science and allow flexibility in the placement of the entry probe. The traditional approach to release the probe ahead of the orbiter may not optimize returned science. In this presentation, we will make the case for mature TPS such as HEEET and PICA. While these materials can enable aerocapture missions, completing the development of conformal PICA and extending Conformal PICA to be more robust, will have significant impact to TPS mass efficiency and significantly enhance science return for future Gas- and Ice-Giant missions.

E Venkatapathy↗