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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1969

Geographic applications program: Reports completed and in preparation

The purpose of the present compilation is to bring together for ready reference a list of all reports, both completed and in preparation, resulting from NASA sponsorship of the geography program. Wherever possible, complete bibliographic citations are given. Since some of the 141 reports listed are not yet completed, citations can not all be in complete form. The reason for including incomplete citations is the sponsor's need for rather rapid feed-back of information so that it can have a bearing on on-going program planning. This should, therefore, be considered a working bibliography.

Alexander, R. H.↗
NASA NTRS1990

Complete the development and construction of a spaceborne hydrogen maser clock

The objective, to complete the development of an engineering model of a spaceborne hydrogen maser, was successfully achieved. A layout of the maser and detail drawings of the physics package was completed during the first 7 months of the contract. A computer model was made for the maser's thermal design. Using numerical computations, heater resistances were established for 7 temperature controlled zones. The physics package includes: a vacuum manifold that houses four sorption pumps capable of scavenging hydrogen for 4 years, a titanium vacuum tank housing the cavity, metallic seals for all vacuum joints, an RF dissociator within the vacuum envelope, a two-layer printed circuit solenoid and four layers of moly-permalloy magnetic shields. Problems were encountered and overcome in the procurements of the PC solenoid and the magnetic shields. After completion of the fabrication of the maser's components, the maser was assembled using these parts and other components made available by SAO, NRL, and NASA from earlier development work. In March, 1990, the vacuum system was assembled, and by May the maser assembly was completed. The magnetic shielding was poor and the shields were removed, reannealed by a local vendor, and the maser was reassembled. The maser began tests in early June and has been oscillating since that time. The test results of the maser are very good and a life test of the maser is being conducted. It is anticipated that the development and construction of a maser to be tested in space under a new contract from NASA's Marshall Space Flight Center will continue.

Vessot, Robert F. C.↗
NASA NTRS1992

Accelerated complete-linearization method for calculating NLTE model stellar atmospheres

Two approaches to accelerating the method of complete linearization for calculating NLTE model stellar atmospheres are suggested. The first one, the so-called Kantorovich variant of the Newton-Raphson method, consists of keeping the Jacobi matrix of the system fixed, which allows us to calculate the costly matrix inversions only a few times and then keep them fixed during the subsequent computations. The second method is an application of the Ng acceleration. Both methods are extremely easy to implement with any model atmosphere code based on complete linearization. It is demonstrated that both methods, and especially their combination, yield a rapidly and globally convergent algorithm, which takes 2 to 5 times less computer time, depending on the model at hand and the required accuracy, than the ordinary complete linearization. Generally, the time gain is more significant for more complicated models. The methods were tested for a broad range of atmospheric parameters, and in all cases they exhibited similar behavior. Ng acceleration applied on the Kantorovich variant thus offers a significant improvement of the standard complete-linearization method, and may now be used for calculating relatively involved NLTE model stellar atmospheres.

Hubeny, I.↗
NASA NTRS2015

Development of a Climatology of Vertically Complete Wind Profiles from Doppler Radar Wind Profiler Systems

This paper describes in detail the QC and splicing methodology for KSC's 50- and 915-MHz DRWP measurements that generates an extensive archive of vertically complete profiles from 0.20-18.45 km. The concurrent POR from each archive extends from April 2000 to December 2009. MSFC NE applies separate but similar QC processes to each of the 50- and 915-MHz DRWP archives. DRWP literature and data examination provide the basis for developing and applying the automated and manual QC processes on both archives. Depending on the month, the QC'ed 50- and 915-MHz DRWP archives retain 52-65% and 16-30% of the possible data, respectively. The 50- and 915-MHz DRWP QC archives retain 84-91% and 85-95%, respectively, of all the available data provided that data exist in the non- QC'ed archives. Next, MSFC NE applies an algorithm to splice concurrent measurements from both DRWP sources. Last, MSFC NE generates a composite profile from the (up to) five available spliced profiles to effectively characterize boundary layer winds and to utilize all possible 915-MHz DRWP measurements at each timestamp. During a given month, roughly 23,000-32,000 complete profiles exist from 0.25-18.45 km from the composite profiles' archive, and approximately 5,000- 27,000 complete profiles exist from an archive utilizing an individual 915-MHz DRWP. One can extract a variety of profile combinations (pairs, triplets, etc.) from this sample for a given application. The sample of vertically complete DRWP wind measurements not only gives launch vehicle customers greater confidence in loads and trajectory assessments versus using balloon output, but also provides flexibility to simulate different DOL situations across applicable altitudes. In addition to increasing sample size and providing more flexibility for DOL simulations in the vehicle design phase, the spliced DRWP database provides any upcoming launch vehicle program with the capability to utilize DRWP profiles on DOL to compute vehicle steering commands, provided the program applies the procedures that this report describes to new DRWP data on DOL. Decker et al. (2015) details how SLS is proposing to use DRWP data and splicing techniques on DOL. Although automation could enhance the current DOL 50-MHz DRWP QC process and could streamline any future DOL 915-MHz DRWP QC and splicing process, the DOL community would still require manual intervention to ensure that the vehicle only uses valid profiles. If a program desires to use high spatial resolution profiles, then the algorithm could randomly add high-frequency components to the DRWP profiles. The spliced DRWP database provides lots of flexibility in how one performs DOL simulations, and the algorithms that this report provides will assist the aerospace and atmospheric communities that are interested in utilizing the DRWP.

Barbre, Robert E., Jr.↗
NASA NTRS2024

Interim Measures Pilot Study Completion Report, South Repeater Building, Solid Waste Management Unit 121, Hydraulic Containment and Groundwater Treatment System for Per- and Polyfluoroalkyl Substances

This Per- and Polyfluoroalkyl Substances (PFAS) Interim Measures Pilot Study Completion Report was prepared for the National Aeronautics and Space Administration by AECOM Technical Services, Inc. under Contract 80KSC019D0010, Task Order 80KSC021F0096. The purpose of this report is to document the pilot study activities at the South Repeater Building, Solid Waste Management Unit (SWMU) 121. This report details the additional assessment activities completed from June 2023 through November 2023 and the subsequent pilot study completed at the South Repeater Building conducted in December 2023 through February 2024. Initial activities were conducted in accordance with the Pilot Study Work Plan (NASA 2023), submitted to the KSC Remediation Team and accepted by the team on October 26, 2023. The objectives of the pilot study were to: • Provide information on aquifer characteristics and to aid in future modeling and remedial design activities to mitigate off-KSC migration of PFAS compounds. • Develop information on characteristics of the surficial aquifer system, specifically transmissivity, storage coefficient, hydraulic conductivity, and vertical hydraulic conductivity. • Obtain data to support construction and calibration of a groundwater flow model. • Acquire design parameters necessary for future remedial design activities, specifically radius of influence, drawdown, flow rates, and pump settings. The following activities were completed to meet the pilot study objectives: • Groundwater and surface water sampling • Geophysical investigation via Hydraulic Profiling Tool (HPT) and soil sampling • Installation of observation and extraction wells via rotosonic techniques • Slug, step, and pump testing activities • Initial modeling efforts for a groundwater hydraulic containment system. Results of the pilot study and modeling activities are being used in the development of a groundwater containment and treatment system for the site.

PFAS↗
DOE OSTI2018

Electronics Upgrades to the Green is Clean Phoswich Detector Systems and Programmatic Implementation at LANL - Phase II Completion - 18403

Los Alamos National Laboratory (LANL) radiological facilities produce low-density room trash that, in many cases, is not contaminated with radioactivity. It has been estimated that 50 to 90% of low-density room trash is free of radioactive contamination and eligible for inclusion in LANL's Green is Clean (GIC) program. The GIC program is a verification program for non-regulated waste from radiation controlled areas that has been actively segregated as nonradioactive through the use of the waste generator's acceptable knowledge. The GIC program has employed two large area Phoswich gamma ray detector systems (HERCULES and ZEUS) to verify that GIC candidate low-density waste contains no radioactivity above background. The large area (127 cm{sup 2}) Phoswich detectors consist of a thin NaI front crystal (3 mm thick) that is optically coupled to a thick CsI back crystal (50 mm thick). The scintillation properties of the two crystals are different enough to allow the system's electronic components to identify the origin of any pulse. The original systems utilize Nuclear Instrumentation Module (NIM) electronics to process the detector signals. The electronics currently in use by the systems are out of date and some components are no longer manufactured. In 2015, phase I upgrades to the GIC detector systems were completed. Phase I objectives included identifying more modern commercially available electronics that could replace the NIM electronics. A pulse shape discriminator (PSD) module, the Digital Gamma Finder (DGF) Pixie-4 made by XIA LLC, that fits in a compact National Instruments{sup TM} PXI chassis was identified to replace the majority of the original NIM bin electronics - a single Pixie-4 module replaces four NIM amplifiers and four NIM pulse shape analyzers (PSAs). Another phase I objective was to show that that new PSD could separate out the NaI and CsI detector signals in a manner equivalent to or better than the NIM electronics. Proof of concept was achieved in 2015 when a prototype upgrade system on ZEUS was built and tested and gating parameter settings were determined that successfully separated the NaI and CsI signals. Phase II required establishing a LabVIEW software interface with Pixie-4 electronics in lieu of the manufacturer provided Pixie-4 software called the Pixie Viewer. In addition, phase II included 1) final optimization of the gating parameters on the prototype Pixie-4 ZEUS system, 2) finalizing Requirements, Design and Test Plan documents, 3) LabVIEW software interface quality assurance (QA) testing, 4) setting up the HERCULES system with the Pixie-4 upgrade electronics, 5) check source and background calibration of both systems, 6) absolute efficiency determination on both systems and 7) procedure revisions and implementation of both systems into routine operations. In 2017 phase II was completed including the preparation of detailed Requirements, Design and Test Plan documents. The LabVIEW interface software that communicates and controls the Pixie-4 has been completed, merged with the preexisting LabVIEW operations software and fully tested. Gating parameter optimization, configuration of HERCULES with the Pixie-4 electronics, check source and background calibrations and absolute efficiency determinations have all been performed - quantitative results are presented in the full paper. For performance validation, comparison measurements of actual GIC bags on both the HERCULES system using the old NIM Bin electronics and the ZEUS system using the upgraded Pixie-4 electronics have been done and evaluated. Finally, procedure revisions and implementation of both systems into routine operations is currently in process and planned to be complete by the end of calendar year 2017. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
NASA NTRS2005

Single visit completeness optimization

The Terrestrial Planet Finder Coronagraph (TPF-C) is a NASA pre-phase. A orbiting telescope with the goal of detecting and characterizing visible and near-IR light refelected from terrestrial planets. We use observational completeness, defined as the fraction of potentially observable planets that are detected, to determine how well the proposed telescope meets the mission goals. Single visit completeness is optimized by slecting the length of observation per target that maximizes the completeness for the ensemble of stars. In this memo we describe the integration time calculations and two approaches to single visit optimization.

optimization↗
DOE OSTI2017

HPC formulations of optimization algorithms for tensor completion

Tensor completion is a powerful tool used to estimate or recover missing values in multi-way data. It has seen great success in domains such as product recommendation and healthcare. Tensor completion is most often accomplished via low-rank sparse tensor factorization, a computationally expensive non-convex optimization problem which has only recently been studied in the context of parallel computing. In this work, we study three optimization algorithms that have been successfully applied to tensor completion: alternating least squares (ALS), stochastic gradient descent (SGD), and coordinate descent (CCD++). We explore opportunities for parallelism on shared- and distributed-memory systems and address challenges such as memory- and operation-efficiency, load balance, cache locality, and communication. Among our advancements are a communication-efficient CCD++ algorithm, an ALS algorithm rich in level-3 BLAS routines, and an SGD algorithm which combines stratification with asynchronous communication. Furthermore, we show that introducing randomization during ALS and CCD++ can accelerate convergence. We evaluate our parallel formulations on a variety of real datasets on a modern supercomputer and demonstrate speedups through 16384 cores. Further, these improvements reduce time-to-solution from hours to seconds on real-world datasets. We show that after our optimizations, ALS is advantageous on parallel systems of small-to-moderate scale, while both ALS and CCD++ provide the lowest time-to-solution on large-scale distributed systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Robust Matrix Completion State Estimation in Distribution Systems

Due to the insufficient measurements in the distribution system state estimation (DSSE), full observability and redundant measurements are difficult to achieve without using the pseudo measurements. The matrix completion state estimation (MCSE) combines the matrix completion and power system model to estimate voltage by exploring the low-rank characteristics of the matrix. This paper proposes a robust matrix completion state estimation (RMCSE) to estimate the voltage in a distribution system under a low-observability condition. Tradition state estimation weighted least squares (WLS) method requires full observability to calculate the states and needs redundant measurements to proceed a bad data detection. The proposed method improves the robustness of the MCSE to bad data by minimizing the rank of the matrix and measurements residual with different weights. It can estimate the system state in a low-observability system and has robust estimates without the bad data detection process in the face of multiple bad data. The method is numerically evaluated on the IEEE 33-node radial distribution system. The estimation performance and robustness of RMCSE are compared with the WLS with the largest normalized residual bad data identification (WLS-LNR), and the MCSE.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2019

Sample Complexity of Power System State Estimation Using Matrix Completion

In this paper, we propose an analytical framework to quantify the amount of data samples needed to obtain accurate state estimation in a power system - a problem known as sample complexity analysis in computer science. Motivated by the increasing adoption of distributed energy resources into the distribution-level grids, it becomes imperative to estimate the state of distribution grids in order to ensure stable operation. Traditional power system state estimation techniques mainly focus on the transmission network which involve solving an overdetermined system and eliminating bad data. However, distribution networks are typically underdetermined due to the large number of connection points and high cost of pervasive installation of measurement devices. In this paper, we consider the recently proposed state-estimation method for underdetermined systems that is based on matrix completion. In particular, a constrained matrix completion algorithm was proposed, wherein the standard matrix completion problem is augmented with additional equality constraints representing the physics (namely power-flow constraints). We analyze the sample complexity of this general method by proving an upper bound on the sample complexity that depends directly on the properties of these constraints that can lower number of needed samples as compared to the unconstrained problem. To demonstrate the improvement that the constraints add to state estimation, we test the method on a 141-bus distribution network case study and compare it to the traditional least squares minimization state estimation method.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Dynamic Matrix Completion Based State Estimation in Distribution Grids

The power distribution network is undergoing tremendous transformation due to an increase in the penetration of renewable energy resources and electric vehicles. These changes have resulted in greater uncertainty and dynamics in the distribution grid states. Therefore, the ability to track and monitor system states has become a critical need for accurate and timely control actions. In this paper, we propose two dynamic sparsity-based state estimation approaches for distribution systems: (1) locally weighted matrix completion (LW-MC) and (2) Bayesian matrix completion with Kalman filter prediction (BMC-KF). The performance of the proposed dynamic state estimation strategies is compared with the classic/static matrix completion (static-MC) approach using the IEEE 37 and IEEE 123 bus test systems. Finally, results indicate that BMC-KF approach outperforms both LW-MC as well as static-MC even when 30% of the measurement data is available. Computational complexity associated with both approaches is quantified.

42 ENGINEERING↗
DOE OSTI2017

Completion Report for Well ER-4-1 Corrective Action Unit 97: Yucca Flat/Climax Mine, Revision 0

Well ER-4-1 was drilled for the U.S. Department of Energy, Nevada National Security Administration Nevada Field Office in support of the Underground Test Area (UGTA) Activity. The well was drilled and completed from March 23 to April 13, 2016, as part of the Corrective Action Investigation Plan (CAIP) for Yucca Flat/Climax Mine Corrective Action Unit (CAU) 97. The primary purpose of the well was to collect hydrogeologic data to assist in validating concepts of the groundwater flow system within the Yucca Flat/Climax Mine CAU, and to test for potential radionuclides in groundwater from the STRAIT (U4a) underground test. The completed well includes one piezometer (p1), to a depth of 663.16 meters (m) (2,175.71 feet [ft]) below ground surface (bgs) and open from the Alluvial aquifer (AA3) to the Oak Spring Butte confining unit (OSBCU) hydrostratigraphic units; and a main completion (m1), which includes 6.625-inch (in.) casing with slotted interval (m1) installed to 906.80 m (2,975.05 ft) bgs in the Lower carbonate aquifer (LCA). A 13.375-in. diameter surface casing was installed from the surface to a depth of 809.00 m (2,654.21 ft) bgs. Well ER-4-1 experienced a number of technical issues during drilling, including borehole instability and sloughing conditions. An intermediate, 10.75-in./9.625-in. casing string was installed to 856.94 m (2,811.48 ft) bgs to control these issues. Borehole stability and erosion problems appear to be associated with the Tunnel Formation (Tn) and the Older tunnel beds (Ton). Overall efforts to stabilize the borehole were successful. Data collected during borehole construction include composite drill cutting samples collected every 3.0 m (10 ft), a partial suite of geophysical logs to a maximum depth of 766.57 m (2,515 ft) bgs, water-quality measurements (including tritium), water-level measurements, and two depth-discrete bailer samples collected at 538.89 m and 646.18 m (1,768 ft and 2,120 ft) bgs respectively. The well penetrated 187.45 m (615 ft) of Quaternary/Tertiary alluvium (QTa), 671.47 m (2,203 ft) of Tertiary Volcanic rocks (Tv), and 66.20 m (217.19 ft) of Paleozoic rocks (|). The stratigraphy and lithology were generally as expected with some exceptions. The top of Paleozoic rocks (|) was predicted to occur at 822.35 m (2,698 ft) bgs and was intercepted at 858.93 m (2,818 ft), a difference of 36.58 m (120 ft). As expected, the Paleozoic rocks (|) are the principal water producing formation in Well ER-4-1. Depth to water was measured after drilling as follows: In the piezometers: p1 at 320.39 m (1,051.16 ft) bgs, (measured January 4, 2017); and in the main production casing interval: m1 at 539.17 m (1,768.92 ft) bgs, (measured December 12, 2016) Geophysical logs and depth-discrete bailer sample analytical results suggest likely zones of prompt injection (underground-test-related) fission products from 472.44 to 481.48 m (1,550 to 1,580 ft) bgs and at approximately 539.50 m (1,770 ft) bgs. Subsequent work at Well ER-4-1 will be included in future reports. Field measurements for tritium were mostly below the Safe Drinking Water Act limit (20,000 picocuries per liter) with the exception of two zones showing elevated tritium concentrations. The first zone is located at approximately 365.76 to 390.14 m (1,200 to 1,280 ft) bgs and a second zone at approximately 542.54 to 566.93 m (1,780 to 1,860 ft) bgs. All Fluid Management Plan requirements were met.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

State Estimation in Low-Observable Distribution Systems Using Matrix Completion: Preprint

The need for distribution system state estimation is on the rise because of the increased penetration of distributed energy resources and flexible load. To manage the distribution systems in real time, operators need to first overcome the challenge of low observability in distribution systems. Also, because of the amount of data present from smart meters, distributed generation measurements, switches, etc., the ideal distribution state estimation methods need to be able to process heterogeneous data. In this paper, an algorithm is developed for voltage phasor estimation in low-observability distribution systems. The algorithm is based on the matrix completion approach from signal processing. The traditional matrix completion formulation is augmented with power-flow constraints to improve results while requiring less data. This method can also use all types of measurements (voltage magnitude, voltage angle, real power, reactive power) to complete the state matrix.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2019

Sample Complexity of Power System State Estimation using Matrix Completion: Preprint

In this paper, we propose an analytical framework to quantify the amount of data samples needed to obtain accurate state estimation in a power system-a problem known as \emph{sample complexity} analysis in computer science. Motivated by the increasing adoption of distributed energy resources into the distribution-level grids, it becomes imperative to estimate the state of distribution grids in order to ensure stable operation. Traditional power system state estimation techniques mainly focus on the transmission network which involve solving an overdetermined system and eliminating bad data. However, distribution networks are typically \emph{underdetermined} due to the large number of connection points and high cost of pervasive installation of measurement devices. In this paper, we consider the recently proposed state-estimation method for underdetermined systems that is based on \emph{matrix completion}. In particular, a \emph{constrained} matrix completion algorithm was proposed, wherein the standard matrix completion problem is augmented with additional equality constraints representing the physics (namely power-flow constraints). We analyze the sample complexity of this general method by proving an upper bound on the sample complexity that depends directly on the properties of these constraints that can lower number of needed samples as compared to the unconstrained problem. To demonstrate the improvement that the constraints add to state estimation, we test the method on a 141-bus distribution network case study and compare it to the traditional least squares minimization state estimation method.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2019

Robust Matrix Completion State Estimation in Distribution Systems: Preprint

Due to the insufficient measurements in the distribution system state estimation (DSSE), full observability and redundant measurements are difficult to achieve without using the pseudo measurements. The matrix completion state estimation (MCSE) combines the matrix completion and power system model to estimate voltage by exploring the low-rank characteristics of the matrix. This paper proposes a robust matrix completion state estimation (RMCSE) to estimate the voltage in a distribution system under a low-observability condition. Tradition state estimation weighted least squares (WLS) method requires full observability to calculate the states and needs redundant measurements to proceed a bad data detection. The proposed method improves the robustness of the MCSE to bad data by minimizing the rank of the matrix and measurements residual with different weights. It can estimate the system state in a low-observability system and has robust estimates without the bad data detection process in the face of multiple bad data. The method is numerically evaluated on the IEEE 33-node radial distribution system. The estimation performance and robustness of RMCSE are compared with the WLS with the largest normalized residual bad data identification (WLS-LNR), and the MCSE.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2025

Modeling Strong Light-Matter Coupling in Correlated Systems: State-Averaged Cavity Quantum Electrodynamics Complete Active Space Self-Consistent Field Theory

The description of strongly correlated systems interacting with quantized cavity modes poses significant theoretical challenges due to the combinatorial scaling of electronic and photonic degrees of freedom. Recent advances addressing this complexity include cavity quantum electrodynamics (QED) generalizations of complete active space configuration interaction and density matrix renormalization group methods. In this work, we introduce a QED extension of state-averaged complete active space self-consistent field theory, which incorporates cavity-induced correlations through a second-order orbital optimization framework with robust convergence properties. The method is implemented using both photon number state and coherent state representations, with the latter showing robust origin invariance in the energies regardless of the completeness of the photonic Fock space. The implementation enables symmetry-free orbital relaxations to account for photon-mediated symmetry breaking in polaritonic systems. Numerical validation on lithium hydride, hydroxide anion, and magnesium hydride cation demonstrates that this method achieves significantly improved accuracy in modeling ground-state and polariton potential energy surfaces compared to QED-CASCI in a fixed orbital basis. In these studies, we reach sub-kcal/mol accuracy in potential energy surface in much smaller active spaces than are required for QED-CASCI. This advancement provides a more robust approach for studying cavity-altered chemical landscapes for ground and exited strongly coupled systems.

CASSCF↗
DOE OSTI2017

Spacetime completeness of non-singular black holes in conformal gravity

We explicitly prove that the Weyl conformal symmetry solves the black hole singularity problem, otherwise unavoidable in a generally covariant local or non-local gravitational theory. Moreover, we yield explicit examples of local and non-local theories enjoying Weyl and diffeomorphism symmetry (in short co-covariant theories). Following the seminal paper by Narlikar and Kembhavi, we provide an explicit construction of singularity-free spherically symmetric and axi-symmetric exact solutions for black hole spacetimes conformally equivalent to the Schwarzschild or the Kerr spacetime. We first check the absence of divergences in the Kretschmann invariant for the rescaled metrics. Afterwords, we show that the new types of black holes are geodesically complete and linked by a Newman-Janis transformation just as in standard general relativity (based on Einstein-Hilbert action). Furthermore, we argue that no massive or massless particles can reach the former Schwarzschild singularity or touch the former Kerr ring singularity in a finite amount of their proper time or of their affine parameter. Finally, we discuss the Raychaudhuri equation in a co-covariant theory and we show that the expansion parameter for congruences of both types of geodesics (for massless and massive particles) never reaches minus infinity. Actually, the null geodesics become parallel at the r =0 point in the Schwarzschild spacetime (the origin) and the focusing of geodesics is avoided. The arguments of regularity of curvature invariants, geodesic completeness, and finiteness of geodesics' expansion parameter ensure us that we are dealing with singularity-free and geodesically-complete black hole spacetimes.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗