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Sparse Linear Algebra Toolkit for Computational Aerodynamics

Finding solutions to sparse linear systems of equations is an essential step in Computational Engineering applications of interest to NASA. Linear systems of equations are composed and solved in almost every computational engineering application. The characteristics of linear systems vary greatly from one application to another. Accordingly, there are a wide variety of methods for the solution of linear systems of equations. The operations and methods prepared by the authors are focused on linear systems of interest to NASA, primarily those associated with Computational Fluid Dynamics (CFD), Aeroelasticity, and Aeroacoustics. The Sparse Linear Algebra Toolkit (SLAT) is a coordinated collection of software featuring operations, methods, and data structures that are useful when solving sparse linear systems of equations on modern computer architectures. The implemented operations and methods are designed and tuned for parallelism in shared memory, in distributed memory, and across the hybrid combination of distributed-shared memory. The toolkit includes novel methods and implementations for modern architectures and facilitates development of new approaches for meeting NASA’s evolving computational engineering challenges using evolving computer architectures that are not available in vendor libraries. In this paper, significant features and interfaces within SLAT are presented and verified for simulations performed with NASA’s CFD solver, FUN3D. The runtime and scaling performance of the Generalized Minimum Residual (GMRES) method implemented in SLAT is analyzed for the linear subproblems within the solution of turbulent Navier-Stokes equations employed in the simulation of high-lift configurations. Prior to this work, the SPARSKIT GMRES implementation was the only Krylov subspace method available within FUN3D. A strong scaling study shows the SLAT GMRES implementation facilitates accurate Reynolds-averaged Navier-Stokes CFD solutions between 15% and 56% faster than the SPARSKIT GMRES implementation.

Stephen L Wood↗
DOE OSTI2021

Influence of Ink Formulation and Drying Conditions on Ionomer Distribution in High-Performance Roll-to-Roll-Coated Gas-Diffusion Electrodes

To enable mass production of fuel cell membrane electrode assemblies (MEAs) catalyst layers production will require continuous roll-to-roll (R2R) coating processes. Gas diffusion electrodes (GDEs) are advantageous for mass production because the catalyst layer can be directely coated on the microporous layer of the gas diffusion media without the need for a decal-transfer process. It is known that the water-to-alcohol ratio in the catalyst ink influences the interactions of the ionomer with the catalyst leading to different distributions of ionomer in spray-coated catalyst layers. It is also known that during drying of colloidal mixtures, like fuel cell inks, factors such as drying rate, particle size, and agglomeration influence how the materials distribute themselves throughout the thickness of the dired film. Thus far there have only been limited studies to understand how process conditions such as ink formulation and drying temperature influence the distribution of ionomer and catalyst coated using scalable methods. This understanding is especially important for GDEs since it is known that having a sufficient amount of ionomer at the catalyst layer-membrane interface is critical for high performance. In this study we have focused on determining how the ratio of water to 1-propanol in the catalyst ink ink and drying temperature influence the distribution of ionomer throughout the thickness of the catalyst layer. Using a combination of Kelvin probe and x-ray photoelectron spectroscopy we show that an ionomer-rich surface is promoted by a higher drying rate and a water-rich catalyst ink. In contrast, a 1-propanol catalyst ink leads to a lower concentration of ionomer on the top surface. Using x-ray computed tomography, we are able to characterize the ionomer distribution throughout the thickness of the layer. We find that, in addition to promoting an ionomer-rich top surface, water-rich inks lead to a more homogenous distribution of ionomer, whereas a 1-propanol-rich ink leads to a more irregular distribution. It is found that MEA performance is improved by selecting conditions and ink formulations that promote ionomer enrichment at the top surface to facilitate a good interface with the membrane. MEAs prepared with a 75 wt% water catalyst ink with a 0.9 I/C have equivalent performance to spray-coated GDEs. Critically, these R2R-coated GDEs do not need an additional ionomer overlayer like the spray-coated GDEs do, reducing the number of processing steps in a manufacturing setting. This work shows that with the appropriate selection of materials, ink formulation, and processing conditions gas-diffusion electrodes are a viable pathway for fuel cell manufacturing.

ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATION↗
DOE OSTI2022

ThunderSecure: deploying real-time intrusion detection for 100G research networks by leveraging stream-based features and one-class classification network

Nowadays, data generated by large-scale scientific experiments are on the scale of petabytes per month. These data are transferred through dedicated high-bandwidth networks (40/100G) across distributed sites for processing, storage, and analysis. Like general purpose networks, research networks experience intrusions. However, monitoring anomalies in such high-speed network traffics is challenging given current cyber-infrastructure. Moreover, traditional network intrusion detection systems (NIDS) are signature based. However, anomaly patterns are difficult to define and that rulesets are often not updated frequently enough to reflect the changes of attack behaviors. We present ThunderSecure, a high-throughput, unsupervised learning-based intrusions detection system for 100G research networks. ThunderSecure implements an efficient packet processing and detection pipeline using multi-cores and GPUs. It extracts statistical and temporal features from real-time network data streams and feeds them to a one-class anomaly detection network. A baseline of normal distribution will be created based on the training observation. Testing traffic deviated from the learned profile will be marked as anomalies. We trained ThunderSecure on hundreds of billions of science data packets mirrored from two 100G network connections at Fermi National Accelerator Laboratory. The detection performance was evaluated on traffic captured from the same research network days and weeks after the training with different types of attack flows injected. Results show that ThunderSecure can recognize science data traffic captured long after the training and made nearly certain detection on the segment of the streams where anomalous flows were injected.

100G research network↗
DOE OSTI2022

Inference for Nonparanormal Partial Correlation via Regularized Rank-Based Nodewise Regression

Abstract Partial correlation is a common tool in studying conditional dependence for Gaussian distributed data. However, partial correlation being zero may not be equivalent to conditional independence under non-Gaussian distributions. In this paper, we propose a statistical inference procedure for partial correlations under the high-dimensional nonparanormal (NPN) model where the observed data are normally distributed after certain monotone transformations. The NPN partial correlation is the partial correlation of the normal transformed data under the NPN model, which is a more general measure of conditional dependence. We estimate the NPN partial correlations by regularized nodewise regression based on the empirical ranks of the original data. A multiple testing procedure is proposed to identify the nonzero NPN partial correlations. The proposed method can be carried out by a simple coordinate descent algorithm for lasso optimization. It is easy-to-implement and computationally more efficient compared to the existing methods for estimating NPN graphical models. Theoretical results are developed to show the asymptotic normality of the proposed estimator and to justify the proposed multiple testing procedure. Numerical simulations and a case study on brain imaging data demonstrate the utility of the proposed procedure and evaluate its performance compared to the existing methods. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Mitigating Cascading Outages in Severe Weather Using Simulation-Based Optimization

Severe weather events can trigger cascading power outages and lead to significant losses. In this work, we investigate cascading outage mitigation under severe weather conditions. Given day-ahead weather forecasts and component failure models, we aim to identify a set of power lines that can be hardened to minimize the expected impact of potential cascading outages. Since the expected load shedding cannot be expressed as an explicit function of line hardening decisions and system states, we developed a cascading outage simulator to estimate the expected value of load shedding under various initial weather-related disruption scenarios generated using a weather forecast. To avoid massive enumeration of all possible combinations of line hardening decisions and reduce the simulation efforts, we employed an efficient simulation-based optimization approach that quickly identifies the (near) optimal line hardening decisions in the presence of both large simulation noises due to the highly variable initial disturbances and system states, and significant randomness in the subsequent cascades. Furthermore, the algorithm is also able to utilize parallel computing to dramatically reduce computation time to support decision making in preparation for severe weather conditions. We performed a case study on the Northeast Power Coordinating Council (NPCC) 140-bus system model to demonstrate that our approach can significantly improve power grid resilience to adverse weather events.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Integrated edge-to-exascale workflow for real-time steering in neutron scattering experiments

We introduce a computational framework that integrates artificial intelligence (AI), machine learning, and high-performance computing to enable real-time steering of neutron scattering experiments using an edge-to-exascale workflow. Focusing on time-of-flight neutron event data at the Spallation Neutron Source, our approach combines temporal processing of four-dimensional neutron event data with predictive modeling for multidimensional crystallography. At the core of this workflow is the Temporal Fusion Transformer model, which provides voxel-level precision in predicting 3D neutron scattering patterns. The system incorporates edge computing for rapid data preprocessing and exascale computing via the Frontier supercomputer for large-scale AI model training, enabling adaptive, data-driven decisions during experiments. This framework optimizes neutron beam time, improves experimental accuracy, and lays the foundation for automation in neutron scattering. Although real-time experiment steering is still in the proof-of-concept stage, the demonstrated potential of this system offers a substantial reduction in data processing time from hours to minutes via distributed training, and significant improvements in model accuracy, setting the stage for widespread adoption across neutron scattering facilities and more efficient exploration of complex material systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

An exploration of online-simulation-driven portfolio scheduling in Workflow Management Systems

Workflow Management Systems used to automate the execution of scientific workflow applications on parallel and distributed computing platforms must make scheduling decisions at runtime. A large number of workflow scheduling algorithms have been proposed in the literature, but often these algorithms are evaluated based on simplifying assumptions that may not hold in practice. Furthermore, published algorithm evaluation and/or comparison results are necessarily only for a subset of all possible scenarios, and thus may not include scenarios relevant to particular use-cases. Consequently, it is difficult for Workflow Management Systems (WMSs) developers to decide which scheduling algorithm should be implemented. To obviate this difficulty, one possible approach is to implement a portfolio of scheduling algorithms and select the most effective algorithm at runtime. One method for performing this selection is to run an online simulation for each algorithm in the portfolio. The algorithm that leads to the best performance, in simulation, is selected for future use. The above simulation-driven portfolio scheduling (SDPS) approach has been proposed in a few parallel and distributed computing contexts. The main objective of this work is to evaluate the feasibility and potential merit of SDPS if implemented in WMSs. Here we perform this evaluation using simulated WMS executions, where the simulations are instantiated from real-world platform and workflow configurations. Our main finding is that SDPS is on par with or outperforms an approach in which a single algorithm is used, where this algorithm is the one that performs best on average across all our experimental scenarios. Furthermore, we find that SDPS remains an attractive proposition even in the presence of high levels of simulation error and for simulators with relatively low levels of sophistication. In many of our experimental scenarios we find that mitigating simulation error at runtime can further improve performance. Finally, we show that simulation overhead can be made sufficiently low for SDPS to be feasible in practice.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Full Scale 3D Computational Model of the Industrial -Scale Coal Fired Boiler Performance for Temperature Sensor Installation Guidance

Abstract Nearly 30% of the electricity is generated by using coal as the primary fuel in the US. One of the major concerns in coal-fired power plants is the failure of boiler tubes that leads to unscheduled maintenance and has a huge economical and societal impact. High temperature flue gas along with ash pass over the boiler tubes, which over time leads to tube failure. Therefore, developing temperature sensors for harsh environments and install them for temperature sensing and boiler tube lifetime prediction is an urgent need. On the side of sensor development, the location of the sensor installation is important for stable sensing performance and easy calibration. In this study, computational fluid dynamics and heat transfer modeling are adopted to establish a full-scale 3-dimensional model of a coal-fired boiler to investigate the flue gas temperature distribution within the boiler and identify the proper locations for sensor installation. We proposed three criteria to select the temperature sensor installation location: (1) select the boiler tube panel away from the sidewalls, (2) select the boiler tube section closer to the top wall of the boiler; and (3) select the boiler tube on the back of the boiler panel (not directly facing the flue gas flow). In these regions, the flue gas temperature is stable, providing an ideal environment for stable temperature sensing and calibration.

Gupta, Tanuj↗
DOE OSTI2016

Exascale Virtualized and Programmable Distributed Cyber Resource Control: Final Scientific Technical Report

Extreme-science drives the need for distributed exascale processing and communications that are carefully, yet flexibly, managed. Exponential growth of data for scientific simulations, experimental data, collaborative data analyses, remote visualization and GRID computing requirements of scientists in fields as diverse as high energy physics, climate change, genomics, fusion, synchrotron radiation, material science, medicine, and other scientific disciplines cannot be accommodated by simply applying existing transport protocols to faster pipes. Further, scientific challenges today demand diverse research teams, heightening the need for and increasing the complexity of collaboration. To address these issues within the network layer and physical layer, we have performed a number of research activities surrounding effective allocation and management of elastic optical network (EON) resources, particularly focusing on FlexGrid transponders. FlexGrid transponders support the opportunity to build Layer-1 connections at a wide range of bandwidths and to reconfigure them rapidly. The new flexibility supports complex new ways of using the physical layer that must be carefully managed and hidden from the scientist end-users. FlexGrid networks utilize flexible (or elastic) spectral bandwidths for each data link without using fixed wavelength grids. The flexibility in spectrum allocation brings many appealing features to network operations. Current networks are designed for the worst case impairments in transmission performance and the assigned spectrum is over-provisioned. In contrast, the FlexGrid networks can operate with the highest spectral efficiency and minimum bandwidth for the given traffic demand while meeting the minimum quality of transmission (QoT) requirement. Two primary focuses of our research are: (1) resource and spectrum allocation (RSA) for IP traffic over EONs, and (2) RSA for cross-domain optical networks. Previous work concentrates primarily on large file transfers within a single domain. Adding support for IP traffic changes the nature of the RSA problem: instead of choosing to accept or deny each request for network support, IP traffic is inherently elastic and thus lends itself to a bandwidth maximization formulation. We developed a number of algorithms that could be easily deployed within existing and new FlexGrid networks, leading to networks that better support scientific collaboration. Cross-domain RSA research is essential to support large-scale FlexGrid networks, since configuration information is generally not shared or coordinated across domains. The results presented here are in their early stages. They are technically feasible and practical, but still require coordination among organizations and equipment owners and a higher-layer framework for managing network requests.

42 ENGINEERING↗
NASA NTRS1991

PDEMOD: Software for control/structures optimization

Because of the possibility of adverse interaction between the control system and the structural dynamics of large, flexible spacecraft, great care must be taken to ensure stability and system performance. Because of the high cost of insertion of mass into low earth orbit, it is prudent to optimize the roles of structure and control systems simultaneously. Because of the difficulty and the computational burden in modeling and analyzing the control structure system dynamics, the total problem is often split and treated iteratively. It would aid design if the control structure system dynamics could be represented in a single system of equations. With the use of the software PDEMOD (Partial Differential Equation Model), it is now possible to optimize structure and control systems simultaneously. The distributed parameter modeling approach enables embedding the control system dynamics into the same equations for the structural dynamics model. By doing this, the current difficulties involved in model order reduction are avoided. The NASA Mini-MAST truss is used an an example for studying integrated control structure design.

Taylor, Lawrence W., Jr.↗
NASA NTRS1992

Comparative study of turbulence models in predicting hypersonic inlet flows

A numerical study was conducted to analyze the performance of different turbulence models when applied to the hypersonic NASA P8 inlet. Computational results from the PARC2D code, which solves the full two-dimensional Reynolds-averaged Navier-Stokes equation, were compared with experimental data. The zero-equation models considered for the study were the Baldwin-Lomax model, the Thomas model, and a combination of the Baldwin-Lomax and Thomas models; the two-equation models considered were the Chien model, the Speziale model (both low Reynolds number), and the Launder and Spalding model (high Reynolds number). The Thomas model performed best among the zero-equation models, and predicted good pressure distributions. The Chien and Speziale models compared very well with the experimental data, and performed better than the Thomas model near the walls.

Kapoor, Kamlesh↗
NASA NTRS1992

Comparative study of turbulence models in predicting hypersonic inlet flows

A numerical study was conducted to analyze the performance of different turbulence models when applied to the hypersonic NASA P8 inlet. Computational results from the PARC2D code, which solves the full two-dimensional Reynolds-averaged Navier-Stokes equation, were compared with experimental data. The zero-equation models considered for the study were the Baldwin-Lomax model, the Thomas model, and a combination of the Baldwin-Lomax and Thomas models; the two-equation models considered were the Chien model, the Speziale model (both low Reynolds number), and the Launder and Spalding model (high Reynolds number). The Thomas model performed best among the zero-equation models, and predicted good pressure distributions. The Chien and Speziale models compared wery well with the experimental data, and performed better than the Thomas model near the walls.

Kapoor, Kamlesh↗
NASA NTRS1993

Three-D Flow Analysis of the Alternate SSME HPOT TAD

This paper describes the results of numerical flow analyses performed in support of design development of the Space Shuttle Main Engine Alternate High Pressure Oxidizer Turbine Turn-around duct (TAD). The flow domain has been modeled using a 3D, Navier-Stokes, general purpose flow solver. The goal of this effort is to achieve an alternate TAD exit flow distribution which closely matches that of the baseline configuration. 3D Navier Stokes CFD analyses were employed to evaluate numerous candidate geometry modifications to the TAD flowpath in order to achieve this goal. The design iterations are summarized, as well as a description of the computational model, numerical results and the conclusions based on these calculations.

Kubinski, Cheryl A.↗
NASA NTRS1996

Concurrent Probabilistic Simulation of High Temperature Composite Structural Response

A computational structural/material analysis and design tool which would meet industry's future demand for expedience and reduced cost is presented. This unique software 'GENOA' is dedicated to parallel and high speed analysis to perform probabilistic evaluation of high temperature composite response of aerospace systems. The development is based on detailed integration and modification of diverse fields of specialized analysis techniques and mathematical models to combine their latest innovative capabilities into a commercially viable software package. The technique is specifically designed to exploit the availability of processors to perform computationally intense probabilistic analysis assessing uncertainties in structural reliability analysis and composite micromechanics. The primary objectives which were achieved in performing the development were: (1) Utilization of the power of parallel processing and static/dynamic load balancing optimization to make the complex simulation of structure, material and processing of high temperature composite affordable; (2) Computational integration and synchronization of probabilistic mathematics, structural/material mechanics and parallel computing; (3) Implementation of an innovative multi-level domain decomposition technique to identify the inherent parallelism, and increasing convergence rates through high- and low-level processor assignment; (4) Creating the framework for Portable Paralleled architecture for the machine independent Multi Instruction Multi Data, (MIMD), Single Instruction Multi Data (SIMD), hybrid and distributed workstation type of computers; and (5) Market evaluation. The results of Phase-2 effort provides a good basis for continuation and warrants Phase-3 government, and industry partnership.

Abdi, Frank↗
NASA NTRS2010

Transient Three-Dimensional Side Load Analysis of Out-of-Round Film Cooled Nozzles

The objective of this study is to investigate the effect of nozzle out-of-roundness on the transient startup side loads at a high altitude, with an anchored computational methodology. The out-of-roundness could be the result of asymmetric loads induced by hardware attached to the nozzle, asymmetric internal stresses induced by previous tests, and deformation, such as creep, from previous tests. The rocket engine studied encompasses a regeneratively cooled thrust chamber and a film cooled nozzle extension with film coolant distributed from a turbine exhaust manifold. The computational methodology is based on an unstructured-grid, pressure-based computational fluid dynamics formulation, and a transient inlet history based on an engine system simulation. Transient startup computations were performed with the out-of-roundness achieved by four different degrees of ovalization: one perfectly round, one slightly out-of-round, one more out-of-round, and one significantly out-of-round. The results show that the separation-line-jump is the peak side load physics for the round, slightly our-of-round, and more out-of-round cases, and the peak side load increases as the degree of out-of-roundness increases. For the significantly out-of-round nozzle, however, the peak side load reduces to comparable to that of the round nozzle and the separation line jump is not the peak side load physics. The counter-intuitive result of the significantly out-of-round case is found to be related to a side force reduction mechanism that splits the effect of the separation-line-jump into two parts, not only in the circumferential direction and most importantly in time.

Wang, Ten-See↗
DOE OSTI2017

Parallel Asynchronous Distributed-Memory Maximal Independent Set Algorithm with Work Ordering

The maximal independent set (MIS) graph problem arises in many applications such as computer vision, information theory, molecular biology, and process scheduling. The growing scale of graph data suggests the use of distributed memory hardware as a cost-effective approach to providing necessary compute and memory resources. Existing distributed memory parallel MIS algorithms rely on synchronous communication and use techniques such as subgraph computations. In this paper, we present an asynchronous distributed-memory parallel graph algorithm that relies on a virtual directed acyclic graph (DAG) that is created during the algorithm execution. We introduce two additional algorithms that save computations by ordering generated work. The first algorithm applies ordering globally to reduce computations, and the second algorithm applies ordering locally at the level of threads to minimize the synchronization overhead. We use two different implementations of Luby’s algorithm variants as baseline to compare the performance of the presented algorithms: (1) vertex-centric Luby A and Luby B implementations, and (2) the CombBLAS linear-algebra Luby A implementation. Results show that proposed algorithms outperform both implementations of Luby algorithms, especially in distributed execution. Furthermore, we show that for low-diameter graphs the algorithm that applies global ordering scales better than other algorithms and for high diameter graphs the original asynchronous algorithm and thread-level ordering algorithm show better performance.

maximal independent set (MIS), distributed memory ↗
DOE OSTI2016

MCNP6 Study of Spatial Fidelity Required for ROC Curve Convergence

The detection of special nuclear material at ports and borders is one of the primary missions of the Department of Homeland Security's (DHS) Domestic Nuclear Detection Office (DNDO). In support of this mission a large-scale modeling and simulation effort has been undertaken with the goal of providing a comprehensive tool-kit that addresses the complexity of problems involved with developing and maintaining technologies deployed to global nuclear detection architecture (GNDA) nodes. An important objective of DNDO's mission is to characterize detector system performance before deployment. In the effort to support this mission, the latest version of MCNP has been used to perform detectability studies of highly enriched uranium (HEU) to populate a database that will be used for the tool-kit. The characterization method used in the implementation of the database is the receiver operating characteristic (ROC). A ROC curve shows the limits of detectability of a given system as well as an overall metric of the efficacy of a detection system. This is accomplished by comparing total detector count distributions to the count distribution produced by background alone. For an in-depth explanation of the theory behind the MCNP production of ROC curves please refer to the paper, 'How To Use The Receiver Operating Characteristic Tally Option in MCNP6'. Building a useful database requires an expansive parameter space. The current tool-kit use nine parameters, which if varied with just 5 options each, results in nearly two million simulations to run. Depending on the set of parameters these simulations can take a few hours or a few weeks to complete. This computational cost necessitates the use of scaling algorithms to interpolate between the data contained in the database. The tool-kit employs a discrete event method to simulate the movement and detector response of special nuclear material (SNM) through a GNDA node. Given the nature of a discrete event simulation, constant motion of SNM must be split up in to discrete spatial detection points. The fidelity required of the spatial splitting to achieve a properly converged answer is investigated here for a neutron detection system. The detector used was a SNAP-3 neutron detector in front of a sphere of SNM in the presence of background radiation. A parameter study has been performed to determine the spatial fidelity required for ROC curve convergence for moving objects in a discrete event simulator. It appears that 5 points is the minimum number of detection points with 7 points needed for one of the parameter sets. The convergence worsens with increases to detection length but appears to improve with increasing detector distance. These results will help future work involving quantifying the accuracy based on given tolerances and parameters. (authors)

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1993

Generic hypersonic vehicle performance model

An integrated computational model of a generic hypersonic vehicle was developed for the purpose of determining the vehicle's performance characteristics, which include the lift, drag, thrust, and moment acting on the vehicle at specified altitude, flight condition, and vehicular configuration. The lift, drag, thrust, and moment are developed for the body fixed coordinate system. These forces and moments arise from both aerodynamic and propulsive sources. SCRAMjet engine performance characteristics, such as fuel flow rate, can also be determined. The vehicle is assumed to be a lifting body with a single aerodynamic control surface. The body shape and control surface location are arbitrary and must be defined. The aerodynamics are calculated using either 2-dimensional Newtonian or modified Newtonian theory and approximate high-Mach-number Prandtl-Meyer expansion theory. Skin-friction drag was also accounted for. The skin-friction drag coefficient is a function of the freestream Mach number. The data for the skin-friction drag coefficient values were taken from NASA Technical Memorandum 102610. The modeling of the vehicle's SCRAMjet engine is based on quasi 1-dimensional gas dynamics for the engine diffuser, nozzle, and the combustor with heat addition. The engine has three variable inputs for control: the engine inlet diffuser area ratio, the total temperature rise through the combustor due to combustion of the fuel, and the engine internal expansion nozzle area ratio. The pressure distribution over the vehicle's lower aft body surface, which acts as an external nozzle, is calculated using a combination of quasi 1-dimensional gas dynamic theory and Newtonian or modified Newtonian theory. The exhaust plume shape is determined by matching the pressure inside the plume, calculated from the gas dynamic equations, with the freestream pressure, calculated from Newtonian or Modified Newtonian theory. In this manner, the pressure distribution along the vehicle after body expansion surface is then determined. The aerodynamic modeling, the engine modeling, and the exhaust plume analysis are described in more detail. A description of the computer code used to perform the above calculations is given and an input/output example is then given. The computer code is available on a Macintosh floppy disk.

Chavez, Frank R.↗