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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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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS2000

A Tool for Automatic Data Distribution for CFD Applications on Structured Grids

Development of HPF versions of NPB and ARC3D has shown that HPF provides an efficient, concise way to express parallelism and to organize data traffic. The use of HPF, as noted in the papers, requires an intimate knowledge of the applications and a detailed analysis of data affinity, data movement, and data granularity. To simplify and accelerate the task of developing HPF versions of existing CFD applications we have designed and implemented ADAPT (Automatic Data Alignment and Placement Tool). ADAPT analyzes a CFD application working on a single structured grid and generates HPF TEMPLATE, (RE)DISTRIBUTION, ALIGNMENT, and INDEPENDENT directives. The directives can be generated on the nest level, subroutine level, application level, or on the application interface level. ADAPT annotates an existing CFD FORTRAN application, performing computations on single or multiple grids. On each grid the application is considered as a sequence of operators, each applied to a set of variables defined in a particular grid domain. ADAPT automatically detects implicit operators (i.e., having data dependences) and explicit operators (without data dependences). For parallelization of an explicit operator ADAPT creates a template for the operator domain, aligns arrays used in the operator with the template, distributes the template, and declares the loops over the distributed dimensions as INDEPENDENT. For parallelization of an implicit operator, the distribution of the operator's domain should be consistent with the operator's dependences. Any dependence between sections distributed on different processors would preclude parallelization if the compiler does not have an ability to pipeline computations. If a data distribution is "orthogonal" to the dependences of an implicit operator, then the loop which implements the operator can be declared as INDEPENDENT. ADAPT starts with an analysis of array index expressions of the loop nests. For each pair of arrays referenced in an assignment statement, it generates an arc in the alignment graph and annotates it with an affinity relation. The template, alignment, and distribution directives for a particular loop nest are then derived from a transitive closure of the affinity relation. A compromise of data distributions in different nests and subroutines is achieved by merging annotated alignment graphs for adjacent nests/stibroutine calls in the nest/call graph of the application in the process called distribution lifting. ADAPT has been implemented as a C++ program running in conjunction with a parallelization tool called CAPTools. ADAPT uses the parse tree, interprocedural analysis and application database generated by CAPTools. It also uses the Directed Graph class, initially implemented in p2d2 (parallel debugger oi distributed programs), and some other classes supporting symbolic computations. ADAPT uses data distribution techniques described. ADAPT was tested with ARC3D and the FT benchmark and has demonstrated a code performance within a factor of 1.5 of handwritten versions.

Frumkin, Michael↗
NASA NTRS2006

Adaptive Modeling, Engineering Analysis and Design of Advanced Aerospace Vehicles

This paper describes initial progress towards the development and enhancement of a set of software tools for rapid adaptive modeling, and conceptual design of advanced aerospace vehicle concepts. With demanding structural and aerodynamic performance requirements, these high fidelity geometry based modeling tools are essential for rapid and accurate engineering analysis at the early concept development stage. This adaptive modeling tool was used for generating vehicle parametric geometry, outer mold line and detailed internal structural layout of wing, fuselage, skin, spars, ribs, control surfaces, frames, bulkheads, floors, etc., that facilitated rapid finite element analysis, sizing study and weight optimization. The high quality outer mold line enabled rapid aerodynamic analysis in order to provide reliable design data at critical flight conditions. Example application for structural design of a conventional aircraft and a high altitude long endurance vehicle configuration are presented. This work was performed under the Conceptual Design Shop sub-project within the Efficient Aerodynamic Shape and Integration project, under the former Vehicle Systems Program. The project objective was to design and assess unconventional atmospheric vehicle concepts efficiently and confidently. The implementation may also dramatically facilitate physics-based systems analysis for the NASA Fundamental Aeronautics Mission. In addition to providing technology for design and development of unconventional aircraft, the techniques for generation of accurate geometry and internal sub-structure and the automated interface with the high fidelity analysis codes could also be applied towards the design of vehicles for the NASA Exploration and Space Science Mission projects.

Mukhopadhyay, Vivek↗
NASA NTRS1994

Merging Intelligent Systems Technologies with CFD Analysis Strategies: Prototype Development at NASA Ames

A collaborative team of researchers from fields of Computational Fluid Dynamics (CFD), fluid physics, computer architectures, and computer science and knowledge engineering have begun work on a prototype system that addresses several of industry's concerns in using NASA-developed CFD codes as part of the design cycle. A major problem exists in the application of CFD technologies within the aeronautics design cycle due primarily to misunderstandings in the ranges of applicability of the various solver codes or turbulence models. Features that arise during the CFD solution process need to be discriminated and recognized as actual flow features with physical support in the geometry and flow conditions of the problem being solved, or as numerical or non-physical errors arising from mis-application of solver code and its parameters, gridding strategies, or discretization. interpolations. The fundamental concept is to develop an intelligent computational system that can accept the engineer's definition of the problem and construct an optimal CFD solution. To do this requires capturing both the knowledge of how to apply the various CFD tools and how to adapt the application of those tools to flow structures as they evolve during the flow simulation. Embedded within this adaptive system approach is the additional desire to automatically identify and quantify the quality of resolution of the pertinent flow structures, be they genuine or error-induced, and then to adjust the solution strategy accordingly. This paper discusses the status of that prototyping effort.

Thompson, David E.↗
NASA NTRS2018

Playbook for UAS: UX of Goal-Oriented Planning & Execution

We are evaluating Playbook for CASAS (Connected Autonomous Smart Aerospace Systems), a tool designed to aid first responders in disaster relief efforts. We are adapting an existing tool, Playbook, to support a future unmanned aircraft system (UAS) swarm demonstration. Playbook for CASAS will be used to plan, edit, and monitor simulated UAS swarms, and we are interested in evaluating the user experience of this prototype as well as developing recommendations for future UAS interfaces. Allocation of roles and responsibilities between human-automation systems is key to promoting productive cooperation between users and automation. Future interfaces, however, must allow for adaptive management of the swarm not a constant split in human-automation control. Our early research indicates that when a single pilot is controlling swarms of robotic agents, such as UAS or ground rovers, operators require a higher level, goal-based interface with usability at its core. Along with that high-level control, users can leverage sensors within the swarm to be notified when lower level actions must be taken by the pilot. First responders working in disaster relief efforts require a high level of situational awareness (SA) and precise control at key moments within a mission. This balance in operator workload paired with SA can lead to improved safety and mission outcomes. Our research below outlines leverage points as well as the balance between human involvement and autonomy in UAS interfaces.

UAS↗
DOE OSTI2014

MO-E-BRD-01: Adapt-A-Thon - Texas Hold’em Invitational

Software tools for image-based adaptive radiotherapy such as deformable image registration, contour propagation and dose mapping have progressed beyond the research setting and are now commercial products available as part of both treatment planning systems and stand-alone applications. These software tools are used together to create clinical workflows to detect, track and evaluate changes in the patient and to accumulate dose. Deviations uncovered in this process are used to guide decisions about replanning/adaptation with the goal of keeping the delivery of prescribed dose “on target” throughout the entire course of radiotherapy. Since the output from one step of the adaptive process is used as an input for another, it is essential to understand and document the uncertainty associated with each of the step and how these uncertainties are propagated. This in turn requires an understanding how the underlying tools work. Unfortunately, important details about the algorithms used to implement these tools are scarce or incomplete, too often for competitive reasons. This is in contrast to the situation involving other basic treatment planning algorithms such as dose calculations, where the medical physics community essentially requires vendors to provide physically important details about their underlying theory and clinical implementation. Vendors should adopt this same level of information sharing when it comes to the tools and techniques for image guided adaptive radiotherapy. The goal of this session is to start this process by inviting vendors and medical physicists to discuss and demonstrate the available tools and describe how they are intended to be used in clinical practice. The format of the session will involve a combination of formal presentations, interactive demonstrations, audience participation and some friendly “Texas style” competition. Learning Objectives: Understand the components of the image-based adaptive radiotherapy process. Understand the how these components are implemented in various commercial systems. Understand the different use cases and workflows currently supported these tools.

60 APPLIED LIFE SCIENCES↗
NASA NTRS2006

Case Study: Test Results of a Tool and Method for In-Flight, Adaptive Control System Verification on a NASA F-15 Flight Research Aircraft

Adaptive control technologies that incorporate learning algorithms have been proposed to enable autonomous flight control and to maintain vehicle performance in the face of unknown, changing, or poorly defined operating environments [1-2]. At the present time, however, it is unknown how adaptive algorithms can be routinely verified, validated, and certified for use in safety-critical applications. Rigorous methods for adaptive software verification end validation must be developed to ensure that. the control software functions as required and is highly safe and reliable. A large gap appears to exist between the point at which control system designers feel the verification process is complete, and when FAA certification officials agree it is complete. Certification of adaptive flight control software verification is complicated by the use of learning algorithms (e.g., neural networks) and degrees of system non-determinism. Of course, analytical efforts must be made in the verification process to place guarantees on learning algorithm stability, rate of convergence, and convergence accuracy. However, to satisfy FAA certification requirements, it must be demonstrated that the adaptive flight control system is also able to fail and still allow the aircraft to be flown safely or to land, while at the same time providing a means of crew notification of the (impending) failure. It was for this purpose that the NASA Ames Confidence Tool was developed [3]. This paper presents the Confidence Tool as a means of providing in-flight software assurance monitoring of an adaptive flight control system. The paper will present the data obtained from flight testing the tool on a specially modified F-15 aircraft designed to simulate loss of flight control faces.

Jacklin, Stephen A.↗
DOE OSTI2025

Fossil Energy in the Hydrogen Economy – A Carbon-Water-Energy Nexus Adaptive Evaluation Platform

This project develops tools to evaluate fossil energy within the hydrogen economy. In particular, the project seeks to explore the Carbon-Water-Energy Nexus and develop adaptive evaluation tools. The primary goal of this project was to evaluate fossil-based hydrogen production methods with an emphasis on their sustainability, particularly with respect to carbon emissions and water use. The overall technical objectives that guided this evaluation consisted of 1) technology assessment, which sought to establish a comprehensive baseline of existing and emerging technologies for the generation, transportation, storage, and end use of hydrogen derived from fossil fuels, 2) carbon footprint analysis through quantification and analysis of carbon emissions associated with these hydrogen technologies, and identification of novel strategies and technologies that can effectively mitigate the carbon footprint, 3) water intensity assessment by evaluating the water needs (water-energy nexus) associated with different fossil fuel-based hydrogen technologies and proposing strategies to reduce water use, thus ensuring sustainable resource management, and 4) review of regulatory, policy and economic trends were examined to provide an overview of the factors influencing the adoption and integration of fossil fuel-derived hydrogen technologies into current energy markets. Beyond the technical outcomes, this project contributed to 5) education and workforce development through targeted training and educational opportunities for engineering students, preparing the next generation of professionals with technical, economic, and regulatory expertise in the hydrogen economy. Ultimately, this effort was intended to train the future workforce to provide useful insights to policymakers, industry stakeholders, and the broader energy community to facilitate informed decisions and responsible development within the hydrogen economy.

08 HYDROGEN↗
DOE OSTI2014

SU-E-T-40: Analysis of Composite MVCT Planning Dosimetry with SBRT of Upper Peripheral Lung Cancer

Purpose: Quantitatively evaluate and compare the final adaptive planning doses of upper peripherally located lung SBRT treated with Tomotherapy using 3rd party software tool. Methods: With tumor located in the upper quadrant of lung, a 3rd party software tool was implemented to evaluate the Tomotherapy composite dosimetry created by adaptive fan beam MVCT images described by RTOG 0915 dose criteria (48 Gy / 4 fractions). The composite doses was then summarized with deformable registration in this package with corresponding target and critical structures. The final dosimetry variation, both for target and critical structures, were evaluated in a tabular format and isodose distribution comparisons. Results: Composite SBRT treatment doses were evaluated with adaptive planning. The PTV and several critical structures were mapped/deformed into the package via DICOM from Tomotherapy after the final composite doses were created. Initial plan versus the final composite plan calculated from verification images were compared. The ITV defined by 4D CT and contoured on MVCT images were correlated in patient repositioning. Final composite dose calculated for PTV coverage has shown 0.1–0.17 cGy coverage (0.2–0.4% of prescription dose) variation. Total lung and cord were both less than 0.17 Gy which represented <0.4% difference. All other critical structure were within statistical significance. The adaptive plans justified/included the breathing and motion during the treatment process. Final 95% isotope line coverage from prescription has been met without issues. Conclusions: With lung tumor location in the upper peripheral area, breathing control was not necessary required during SBRT treatment using Tomotherapy technique. Slow fan beam CT provides definitive ITV information and the adaptive composite plan for all fractions were suitable for final dose delivery. The final composite dose calculated with Tomotherapy adaptive tool indicated that the composite dosimetry justified the target location with SBRT delivery, safe with minimum margin of errors.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2024

Using scalable computer vision to automate high-throughput semiconductor characterization

Abstract High-throughput materials synthesis methods, crucial for discovering novel functional materials, face a bottleneck in property characterization. These high-throughput synthesis tools produce 10 4 samples per hour using ink-based deposition while most characterization methods are either slow (conventional rates of 10 1 samples per hour) or rigid (e.g., designed for standard thin films), resulting in a bottleneck. To address this, we propose automated characterization (autocharacterization) tools that leverage adaptive computer vision for an 85x faster throughput compared to non-automated workflows. Our tools include a generalizable composition mapping tool and two scalable autocharacterization algorithms that: (1) autonomously compute the band gaps of 200 compositions in 6 minutes, and (2) autonomously compute the environmental stability of 200 compositions in 20 minutes, achieving 98.5% and 96.9% accuracy, respectively, when benchmarked against domain expert manual evaluation. These tools, demonstrated on the formamidinium (FA) and methylammonium (MA) mixed-cation perovskite system FA 1−x MA x PbI 3 , 0 ≤ x ≤ 1, significantly accelerate the characterization process, synchronizing it closer to the rate of high-throughput synthesis.

Science & Technology - Other Topics↗
NASA NTRSDate not supplied

Mission Design Strategies for Rendezvous and Servicing of Sun-Earth Libration Point Missions

With the launch of the James Webb Space Telescope (JWST) and future launches of the Roman Space Telescope (RST) and larger telescopes such as the proposed Habitable Worlds Observatory (HWO), the questions of where and how to rendezvous to enable servicing of these telescope missions arise. To aid in determining the allowable locations to rendezvous for servicing from a trajectory design approach, our previous research and analysis has shown feasible transfer trajectories between the Sun-Earth Libration L2 region (Quasi-Halo orbit) and the Earth Moon vicinity (Distant Retrograde Orbit (DRO), Quasi-Halo Orbit, Halo Orbit, and Near Rectilinear Halo Orbit (NRHO)), along with the related transfer constraints and the corresponding total fuel mass costs. We now focus on operational-like optimal scenarios for these transfers to complete a rendezvous to permit servicing at various Libration orbit locations. In this paper, we address how operational navigation and maneuver execution uncertainties impact the rendezvous timing and fuel mass (via ΔV). While optimization techniques are applied to ensure minimal ΔVs, we also analyze various transfer trajectory durations and rendezvous arrival geometries. The analysis presented includes dynamical system approaches and applies optimization through several tools including the Adaptive Trajectory Design (ATD) module as an initial guess, numerical computation using the General Mission Analysis Tool (GMAT) and Systems Tool Kit (STK) with higher fidelity perturbation modeling, and recently developed optimization tools that incorporate dynamical systems directly into the optimization process. Transfer trajectory options examined include a direct transfer from Earth versus departures from the previously examined Earth-Moon regime orbits, e.g. NRHOs and DROs. A return from Sun-Earth to lunar vicinity for a complete servicing after a rendezvous in Sun-Earth orbit is also considered. The resultant ΔV’s for each scenario is provided with discussions on various transfer trades, rendezvous considerations, and orbital limitations from the dynamical systems. A Poincare-like mapping of trajectory and rendezvous conditions categorize feasible and optimum transfer approaches, dependent on Sun-Earth orbit mission parameters. The focus of this research addresses questions for upcoming trades regarding servicing options for current and future Sun-Earth L1 and L2 missions. By doing a comprehensive analysis approach (dynamical systems + high fidelity optimization with true operational constraints), this paper will serve as a guide for mission architecture and operational trade considerations.

Maneuver Design↗
DOE OSTI2023

PathTracer Comprehensively Identifies Hypoxia-Induced Dormancy Adaptations in Mycobacterium tuberculosis

Mining large-scale data to discover biologically relevant information remains a challenge despite the rapid development of bioinformatics tools. Here, we have developed a new tool, PathTracer, to identify biologically relevant information flows by mining genome-wide protein–protein interaction networks following integration of gene expression data. PathTracer successfully mines interactions between genes and traces the most perturbed paths of perceived activities under the conditions of the study. Here, we further demonstrated the utility of this tool by identifying adaptation mechanisms of hypoxia-induced dormancy in Mycobacterium tuberculosis (Mtb).

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2016

SU-F-J-98: Improvement and Evaluation Of Deformation Image Registration On Parotid Glands During Radiation Therapy for Nasopharyngeal Cancer

Purpose: To quantitatively evaluate the strategic innovation and accuracy variation of deformation registration algorithm for parotid glands using the similarity Dice coefficient during the course of radiation therapy (RT) for nasopharyngeal cancer (NPC). Methods: Daily MVCT data for 10 patients with pathologically proven nasopharyngeal cancers were analyzed. The data were acquired using tomotherapy (TomoTherapy, Accuray) at the PLA General Hospital. The prescription dose to the primary target was 70Gy in 33 fractions. Two kinds of contours for parotid glands on daily MVCTs were obtained by populating these contours from planning CTs to the daily CTs via rigid-body registration with or without the rotation shifts using the in-house tools and the Adaptive plan software (Adaptive Plan, TomoTherapy), and were edited manually if necessary. The diffeomorphic Demons algorithm developed in the in-house tool was used to propagate the parotid structures from the daily CTs to planning CTs. The differences of the mapped parotid contours in two methods were evaluated using Dice similarity index (DSI). Two-tailed t-test analysis was carried out to compare the DSI changes during the course of RT. Results: For 10 patient plans, the accuracy of deformation image registration (DIR) with the rotation shift was obviously better than those without the rotation shift. The Dice scores of the ipsi- and contra-lateral parotids for with and without the rotation shifts were found to be correlated with each other [0.904±0.031 vs 0.919±0.030 (p<0.001); 0.900±0.031 vs 0.910±0.032 (p<0.001)]. The Dice scores for the parotids have shown the reduction with the changes of parotid volumes during RT. The DSI values between the first and last fraction were 0.932±0.020 vs 0.899±0.030 in 10 patient plans. Conclusion: DIR was successfully improved using the strategic innovation for ART. And the decrease of DIR accuracy has also been found during the delivery of fractionated radiotherapy. This work was supported in part by the grant from Chinese Natural Science Foundation (Grant No. 11105225).

60 APPLIED LIFE SCIENCES↗
DOE OSTI2024

Ad-Mat: Adaptations of Mature Manufacturing Strategies for Accelerated Redox Flow Battery Deployment

The concept of the Ad-Mat approach is to leverage existing adjacent markets across a broad scope of technologies in order to reduce the manufacturing learning curve and ultimately accelerate redox flow battery (RFB) deployment at scale. Lithium-ion batteries (LIBs) are currently the dominant energy storage technology, and they came to technological maturity under unique market conditions when there was no meaningful competition in the consumer electronic and electric vehicle (EV) space. Today, alternative chemistries that may be technologically better-suited for long-duration storage applications are experiencing a high barrier to entry. This is in large part due to the substantial bias towards the scaled-up production and supply chain that now exists for LIBs. In the case of RFBs in particular, numerous analyses have suggested that RFBs should theoretically have a much lower system cost than LIBs - however, this relies on a mature and competitive manufacturing landscape, which has been extremely challenging to achieve for both flow batteries and other LIB competitors. At the moment, LIB alternatives tend to have isolated small-scale manufacturing pathways, which preclude the economies of scale that would be required to compete with the mature LIB industry. In the present state of the industry, niche manufacturing tools and approaches have evolved to support each alternative technology, such that there is substantial replication and duplication in effort. Continuing to pursue a strategy of isolated manufacturing processes/approaches for each LIB-alternative may never allow for at-scale deployment. In order for RFBs to meaningfully compete with LIBs in the realm of LDES, a new disruptive approach based on cross-industry learning and coordination is needed - and this is exactly what our Ad-Mat concept aims to tackle. In this re-envisioned manufacturing landscape, tools and processes from mature industries can be adapted and deployed across the range of alternative energy storage technologies. Adapting tools, equipment, processes, and industrial learning from mature industries to meet the technological requirements of RFBs would open new markets for existing OEMs in adjacent industries, would prevent unnecessary duplication and re-development, would improve efficiency across the manufacturing chain, and would ultimately support reduced costs and accelerated deployment of RFBs at scale.

adaptive manufacturing↗
DOE OSTI2023

Predicting future well performance for environmental remediation design using deep learning

Here in this study, we developed a deep learning (DL) framework with a multi-channel three-dimensional convolutional neural network (MC3D-CNN) to predict well performance and thereby assist future environmental remediation design. Such prediction of extraction well performance at designated locations is critical for configuring pump-and-treat (P&T) well network design and operation, setting reasonable target closure dates for overall remedying, and estimating remedy costs. The framework is developed with operational and monitoring data routinely collected during P&T remedy operations, including well extraction and injection rates as well as in situ contaminant concentrations. Traditionally, the collected data were rarely used for purposes other than assessing past well performance and the accuracy of the conceptual site model. However, recent advances in data-driven computational approaches enable better use of the large datasets to inform future well performance, enhance site characterization, and improve remediation planning. In this study, we established a DL framework to integrate transient three-dimensional contaminant plumes and multiple aquifer properties (e.g., hydraulic conductivity and hydrostratigraphic maps) to identify characteristic patterns controlling and representing extraction well mass recovery, aiming at providing future mass recovery estimates for existing wells and candidate wells at any proposed locations. We evaluated our framework by using a realistic synthetic dataset generated from a well-calibrated flow and transport model used in the 200 West Area of the U.S. Department of Energy’s Hanford Site in southeastern Washington state. The multi-channel feature in our framework allows integration of various types and temporal densities of training datasets for DL model development. Overall, we found that the trained DL model achieved an accuracy of over 90% in ranking extraction well performance in validation datasets, and over 80% in predicting high-performance-ranking well locations. This data-informed approach provides a flexible tool to support adaptive site management, streamline decision-making, and potentially reduce remediation time and costs. Our DL framework can be used as a filtering tool to improve the current P&T network optimization design by reducing the number of candidate well locations.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

RANGE: A robust adaptive nature-inspired global explorer of potential energy surfaces

With the growing demand for realistic representations of chemical structures and the advent of exascale computing, the intelligent sampling of potential energy surfaces and efficient identification of global minima have become more essential but also more feasible. Building on prior studies demonstrating the efficiency of the Artificial Bee Colony (ABC) swarm intelligence algorithm, we report a hybrid metaheuristic framework that integrates the adaptive exploration capabilities of ABC coupled with the exploitation strengths of genetic algorithms (GA) in a scalable, Python-based implementation. The resulting tool, RANGE (Robust Adaptive Nature-inspired Global Explorer), provides seamless interfaces to multiple potential energy evaluators, either directly or via widely used Python libraries, and is designed for high-performance computing environments. We describe the implementation details of RANGE and evaluate its performance, relative to ABC- or GA-alone based algorithms, on a variety of chemical systems, including molecular clusters and heterogeneous surfaces. In conclusion, our results demonstrate RANGE’s efficiency, robustness, and broad applicability in addressing challenging global optimization problems in computational chemistry and materials science.

Algorithms and data structure↗
NASA NTRS2011

The Creation of a CPU Timer for High Fidelity Programs

Using C and C++ programming languages, a tool was developed that measures the efficiency of a program by recording the amount of CPU time that various functions consume. By inserting the tool between lines of code in the program, one can receive a detailed report of the absolute and relative time consumption associated with each section. After adapting the generic tool for a high-fidelity launch vehicle simulation program called MAVERIC, the components of a frequently used function called "derivatives ( )" were measured. Out of the 34 sub-functions in "derivatives ( )", it was found that the top 8 sub-functions made up 83.1% of the total time spent. In order to decrease the overall run time of MAVERIC, a launch vehicle simulation program, a change was implemented in the sub-function "Event_Controller ( )". Reformatting "Event_Controller ( )" led to a 36.9% decrease in the total CPU time spent by that sub-function, and a 3.2% decrease in the total CPU time spent by the overarching function "derivatives ( )".

Dick, Aidan A.↗