Search NASA⌕ Search

SEARCH · Search NASA

Results for “computation fluid dynamics”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

A study on computational fluid dynamics modeling of a refrigerated container for COVID-19 vaccine distribution with experimental validation

A key issue with the distribution of vaccines to prevent COVID-19 is the temperature level required during transport, storage, and distribution. Typical refrigerated transport containers can provide a temperature-controlled environment down to -30 °C. However, the Pfizer vaccine must be carefully transported and stored under a lower temperature between -80 °C and -60 °C. One way to provide the required temperature is to pack the vaccine vials into small packages containing dry ice. Dry ice sublimates from a solid to a gas, which limits the allowable transport duration. This can be mitigated by transporting in a -30 °C refrigerated container. Moreover, because the dry ice will sublimate and thereby release CO 2 gas into the transport container, monitoring the CO 2 concentration within the refrigerated container is also essential. In the present work, a 3D computational fluid dynamics model was developed based on a commercially available refrigerated container and validated with experimental data. The airflow, temperature distribution, and CO 2 concentration within the container were obtained from the simulations. The modeling results can provide guidance on preparing experimental setups, thus saving time and lowering cost, and also provide insight into safety precautions needed to avoid hazardous conditions associated with the release of CO 2 during vaccine distribution.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS1997

Computational Fluid Dynamics Analysis Method Developed for Rocket-Based Combined Cycle Engine Inlet

Renewed interest in hypersonic propulsion systems has led to research programs investigating combined cycle engines that are designed to operate efficiently across the flight regime. The Rocket-Based Combined Cycle Engine is a propulsion system under development at the NASA Lewis Research Center. This engine integrates a high specific impulse, low thrust-to-weight, airbreathing engine with a low-impulse, high thrust-to-weight rocket. From takeoff to Mach 2.5, the engine operates as an air-augmented rocket. At Mach 2.5, the engine becomes a dual-mode ramjet; and beyond Mach 8, the rocket is turned back on. One Rocket-Based Combined Cycle Engine variation known as the "Strut-Jet" concept is being investigated jointly by NASA Lewis, the U.S. Air Force, Gencorp Aerojet, General Applied Science Labs (GASL), and Lockheed Martin Corporation. Work thus far has included wind tunnel experiments and computational fluid dynamics (CFD) investigations with the NPARC code. The CFD method was initiated by modeling the geometry of the Strut-Jet with the GRIDGEN structured grid generator. Grids representing a subscale inlet model and the full-scale demonstrator geometry were constructed. These grids modeled one-half of the symmetric inlet flow path, including the precompression plate, diverter, center duct, side duct, and combustor. After the grid generation, full Navier-Stokes flow simulations were conducted with the NPARC Navier-Stokes code. The Chien low-Reynolds-number k-e turbulence model was employed to simulate the high-speed turbulent flow. Finally, the CFD solutions were postprocessed with a Fortran code. This code provided wall static pressure distributions, pitot pressure distributions, mass flow rates, and internal drag. These results were compared with experimental data from a subscale inlet test for code validation; then they were used to help evaluate the demonstrator engine net thrust.

Source record↗
NASA NTRS2015

Computational Fluid Dynamics Simulation of Dual Bell Nozzle Film Cooling

Marshall Space Flight Center (MSFC) and Armstrong Flight Research Center (AFRC) are working together to advance the technology readiness level (TRL) of the dual bell nozzle concept. Dual bell nozzles are a form of altitude compensating nozzle that consists of two connecting bell contours. At low altitude the nozzle flows fully in the first, relatively lower area ratio, nozzle. The nozzle flow separates from the wall at the inflection point which joins the two bell contours. This relatively low expansion results in higher nozzle efficiency during the low altitude portion of the launch. As ambient pressure decreases with increasing altitude, the nozzle flow will expand to fill the relatively large area ratio second nozzle. The larger area ratio of the second bell enables higher Isp during the high altitude and vacuum portions of the launch. Despite a long history of theoretical consideration and promise towards improving rocket performance, dual bell nozzles have yet to be developed for practical use and have seen only limited testing. One barrier to use of dual bell nozzles is the lack of control over the nozzle flow transition from the first bell to the second bell during operation. A method that this team is pursuing to enhance the controllability of the nozzle flow transition is manipulation of the film coolant that is injected near the inflection between the two bell contours. Computational fluid dynamics (CFD) analysis is being run to assess the degree of control over nozzle flow transition generated via manipulation of the film injection. A cold flow dual bell nozzle, without film coolant, was tested over a range of simulated altitudes in 2004 in MSFC's nozzle test facility. Both NASA centers have performed a series of simulations of that dual bell to validate their computational models. Those CFD results are compared to the experimental results within this paper. MSFC then proceeded to add film injection to the CFD grid of the dual bell nozzle. A series of nozzle pressure ratios and film coolant flow rates are investigated to determine the effect of the film injection on the nozzle flow transition behavior. The results of this CFD study of a dual bell with film injection are presented in this paper.

Braman, Kalen↗
DOE OSTI2026

3D probabilistic fracture mechanics / computational fluid dynamics simulation of a reactor pressure vessel under transient conditions

Reactor pressure vessels (RPVs) are safety-critical light-water-reactor components that, under irradiation, experience long-term material degradation in the form of embrittlement. This can increase their susceptibility to fracture under thermal-shock conditions, which could occur during off-normal transients such as loss-of-coolant accidents (LOCAs). During a LOCA, the most severe conditions for the RPV occur when emergency core cooling water is injected through the cold legs into the water-and-steam-filled RPV. The rapid cooling of the downcomer and internal RPV surface causes decreased temperature and elevated thermally driven tensile stresses in the RPV wall. This, combined with long-term material embrittlement, may cause fracture initiation at pre-existing flaws, challenging the integrity of the RPV. Assessing RPV integrity during transients with a large spatial variation in the coolant temperature requires a modeling approach that considers the effects of spatially varying coolant temperature on the fracture probability of a population of flaws distributed throughout the RPV, accounting for spatially varying embrittlement. Here, the present study addresses this need by demonstrating first-of-its-kind coupling of a high-fidelity 3D computational fluid dynamics code with 3D probabilistic fracture mechanics This was accomplished using representative models of a pressurized-water reactor subjected to small- and medium-break LOCA conditions, both of which can result in large spatial temperature variations. While the observed impact of accounting for 3D effects was minimal under the small-break LOCA this study indicates a significant increase in the probability of fracture initiation under the medium-break LOCA when 3D effects are considered, relative to a spatially uniform cooling scenario.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

Engine Combustion System Optimization Using Computational Fluid Dynamics and Machine Learning: A Methodological Approach

Gasoline compression ignition (GCI) engines are considered an attractive alternative to traditional spark-ignition and diesel engines. Here, a Machine Learning-Grid Gradient Ascent (ML-GGA) approach was developed to optimize the performance of internal combustion engines. ML offers a pathway to transform complex physical processes that occur in a combustion engine into compact informational processes. The developed ML-GGA model was compared with a recently developed Machine Learning-Genetic Algorithm (ML-GA). Detailed investigations of optimization solver parameters and variable limit extension were performed in the present ML-GGA model to improve the accuracy and robustness of the optimization process. Detailed descriptions of the different procedures, optimization tools, and criteria that must be followed for a successful output are provided here. The developed ML-GGA approach was used to optimize the operating conditions (case 1) and the piston bowl design (case 2) of a heavy-duty diesel engine running on a gasoline fuel with a research octane number (RON) of 80. The ML-GGA approach yielded >2% improvements in the merit function, compared with the optimum obtained from a thorough computational fluid dynamics (CFD) guided system optimization. The predictions from the ML-GGA approach were validated with engine CFD simulations. This study demonstrates the potential of ML-GGA to significantly reduce the time needed for optimization problems, without loss in accuracy compared with traditional approaches.

33 ADVANCED PROPULSION SYSTEMS↗
NASA NTRSDate not supplied

Computational Fluid Dynamics Simulation of Methane Slosh and Drain Experiments

NASA possesses a wealth of historical cryogenic experiments that provide valuable insights into the design, troubleshooting, and understanding involved in the complex fluid and thermodynamics of managing cryogenic propellants. One approach to leveraging these historical datasets is by simulating these experiments to validate the accuracy of simulation environments and constituent models. This study focused on simulating a selection of the K site test series for both static and sloshing pressurized liquid methane draining experiments conducted at NASA in the 1970’s, utilizing computational fluid dynamics. The simulations were performed in the ANSYS FLUENT environment using the Volume of Fluid (VOF) numerical approach. A k-omega turbulence model was used with interfacial turbulence damping, and accurately predicted the amount of pressurant needed to maintain the required tank pressure throughout the static drain. The static simulation predicted the temperature stratification in the ullage observed at the end of the drain. During the methane expulsion with sloshing test, many features were successfully captured using the k-omega turbulence model with interfacial turbulence damping included. The rate of phase change and liquid temperature was overpredicted compared to the experimental measurements. The overprediction may be attributed to uncertainties in the vessel geometry, methane pressurant temperature and composition, and methodological differences in how the sloshing frequency was adjusted during the expulsion.

Cryogenic Propellants↗
DOE OSTI2024

Isothermal Compressor Computational Fluid Dynamics Simulations (Final Report)

Carnot Compression is a startup company developing an innovative technology for air and gas compression. This technology is inherently oil-free and isothermal due to the use of water to simultaneously compress and cool the gas throughout the process. Isothermal gas compression eliminates the need to cool the compressed gas so it may lead to significant energy savings. The development of Carnot’s isothermal compressor is limited by the lack of insight into the flow and detailed behavior of the fluids (water and air/gas) inside the air end. Previous and current attempts at Computational Fluid Dynamics (CFD) simulations by Carnot have been unable to provide the level of accuracy required to use simulations for technology development. In this Cooperative Research and Development Agreement (CRADA) project, Oak Ridge National Laboratory (ORNL) used the CFD package Star-CCM+ to successfully develop a CFD simulation, providing the much-needed insight required to speed up the development. The work is intended to enable Carnot Compression to unlock its technology potential to a level sufficient for commercialization of the intended first product. The work will also prepare Carnot to scale the technology to much larger and more energy intensive applications, positioning it to broaden the product applications. Successful development of the technology has the potential to result in 20% or more efficiency gains for compression processes, or about $3 billion in annual energy savings potential for the U.S. alone.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
NASA NTRS2015

Characterization of the Scale Model Acoustic Test Overpressure Environment using Computational Fluid Dynamics

The Scale Model Acoustic Test (SMAT) is a 5% scale test of the Space Launch System (SLS), which is currently being designed at Marshall Space Flight Center (MSFC). The purpose of this test is to characterize and understand a variety of acoustic phenomena that occur during the early portions of lift off, one being the overpressure environment that develops shortly after booster ignition. The pressure waves that propagate from the mobile launcher (ML) exhaust hole are defined as the ignition overpressure (IOP), while the portion of the pressure waves that exit the duct or trench are the duct overpressure (DOP). Distinguishing the IOP and DOP in scale model test data has been difficult in past experiences and in early SMAT results, due to the effects of scaling the geometry. The speed of sound of the air and combustion gas constituents is not scaled, and therefore the SMAT pressure waves propagate at approximately the same speed as occurs in full scale. However, the SMAT geometry is twenty times smaller, allowing the pressure waves to move down the exhaust hole, through the trench and duct, and impact the vehicle model much faster than occurs at full scale. The DOP waves impact portions of the vehicle at the same time as the IOP waves, making it difficult to distinguish the different waves and fully understand the data. To better understand the SMAT data, a computational fluid dynamics (CFD) analysis was performed with a fictitious geometry that isolates the IOP and DOP. The upper and lower portions of the domain were segregated to accomplish the isolation in such a way that the flow physics were not significantly altered. The Loci/CHEM CFD software program was used to perform this analysis.

Nielsen, Tanner↗
DOE OSTI2022

Computational Fluid Dynamics and Heat Transfer Modeling of a Dimpled Heat Exchanger

Multiphysics and analytical calculations were conducted for a heat exchanger with passive, natural circulation flow. A glycol/water working fluid convects the heat to a dimpled heat exchanger shell, which subsequently transfers the heat to the soil, which acts as the ultimate heat sink. Because the system is fully-passive, it is not subject to the expenses, maintenance, and mechanical breakdowns associated with moving parts. Density, heat capacity, and thermal conductivity material properties were measured for various soil samples, and subsequently included as input for the soil heat conduction model. The soil model was coupled to a computational fluid dynamics (CFD) heat exchanger model that included the dynamic Smagorinsky large eddy simulation and k- omega turbulence models. The analysis showed that the fluid dynamics and heat transfer models worked properly, albeit at a slow pace. Nevertheless, the coupled CFD/heat conduction simulation ran long enough to determine a key parameter—the amount of heat conducted from the heat exchanger to the ground. This unique performance value, along with experimental data, was used as input for stand-alone, fast-running CFD models, as well as boundaries to obtain solutions to partial differential equations for soil heat conduction.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

Computational Fluid Dynamics Simulations of Glass Vitrification Refractory Coupon Tests

The Waste Treatment and Immobilization Plant (WTP) at the Hanford site is nearing the start of the Direct-Feed Low-Activity Waste (DFLAW) operations. DFLAW is destined to convert a pretreated low activity waste portion of the 56 million gallons of tank waste into a stable solid glass. In the subsequent decade completion of the high-level waste (HLW) facility is anticipated. Sustained operational missions of both LAW and HLW melter facilities are expected over multiple decades. In high-temperature glass melters, the refractory lining corrodes over time, which could potentially be an issue for longer term operations, this refractory corrosion is higher at the level of the glass-air interface due to surface tension driven flow. The glass viscosity, melt pool temperature, and glass chemical composition can impact the rate at which the refractory corrodes. This rate is important to quantify for the various waste glasses to be produced at the WTP since the integrity of the refractory should not be a limiting factor affecting the lifetime of the melter. To this end, a series of glasses representative of the first batches of waste glass produced by the WTP will be melted in small-scale crucibles with Monofrax® K-3 coupons inserted. The corrosion of the K-3 will be measured in the melt and at the meltline (or neckline). A model for the corrosion rate will be constructed and implemented into a previously developed framework for a computational fluid dynamics (CFD) model of the full-scale WTP. To assist with experimental design and validate the implementation of the model in the full-scale melter, CFD simulations of the small-scale crucible tests were performed. The bubbling that occurs in the small-scale crucible is initially validated here with a model that uses silicone oil at room temperature. The viscosity of the oil ranges from 1 to 100 Pa•s, which corresponds to operating glass pool temperatures near 1150 °C down to idling temperatures near 950 °C. The simulation results show good agreement with the bubble sizes that form during experiments. CFD modeling of the crucible setup was used to determine bubbling characteristics to match the range of near-wall velocities expected in the full-scale WTP. This study presents the initial CFD modeling results, corrosion testing plan, and some preliminary corrosion samples with an outline for the next steps for the development of the corrosion model.

Abboud, Alexander W. [Idaho National Lab]↗
NASA NTRS2015

Computational Fluid Dynamics Study on the Effects of RATO Timing on the Scale Model Acoustic Test

The Scale Model Acoustic Test (SMAT) is a 5% scale test of the Space Launch System (SLS), which is currently being designed at Marshall Space Flight Center (MSFC). The purpose of this test is to characterize and understand a variety of acoustic phenomena that occur during the early portions of lift off, one being the overpressure environment that develops shortly after booster ignition. The SLS lift off configuration consists of four RS-25 liquid thrusters on the core stage, with two solid boosters connected to each side. Past experience with scale model testing at MSFC (in ER42), has shown that there is a delay in the ignition of the Rocket Assisted Take Off (RATO) motor, which is used as the 5% scale analog of the solid boosters, after the signal to ignite is given. This delay can range from 0 to 16.5ms. While this small of a delay maybe insignificant in the case of the full scale SLS, it can significantly alter the data obtained during the SMAT due to the much smaller geometry. The speed of sound of the air and combustion gas constituents is not scaled, and therefore the SMAT pressure waves propagate at approximately the same speed as occurs during full scale. However, the SMAT geometry is much smaller allowing the pressure waves to move down the exhaust duct, through the trench, and impact the vehicle model much faster than occurs at full scale. To better understand the effect of the RATO timing simultaneity on the SMAT IOP test data, a computational fluid dynamics (CFD) analysis was performed using the Loci/CHEM CFD software program. Five different timing offsets, based on RATO ignition delay statistics, were simulated. A variety of results and comparisons will be given, assessing the overall effect of RATO timing simultaneity on the SMAT overpressure environment.

Nielsen, Tanner↗
DOE OSTI2021

Computational Fluid Dynamics (CFD) Simulations of Taylor Bubbles in Vertical and Inclined Pipes with Upward and Downward Liquid Flow

Summary Two-phase flow is a common occurrence in pipes of oil and gas developments. Current predictive tools are based on the mechanistic two-fluid model, which requires the use of closure relations to predict integral flow parameters such as liquid holdup (or void fraction) and pressure gradient. However, these closure relations carry the highest uncertainties in the model. In particular, significant discrepancies have been found between experimental data and closure relations for the Taylor bubble velocity in slug flow, which has been determined to strongly affect the mechanistic model predictions (Lizarraga-García 2016). In this work, we study the behavior of Taylor bubbles in vertical and inclined pipes with upward and downward flow using a validated 3D computational fluid dynamics (CFD) approach with level set method implemented in a commercial code. A total of 56 cases are simulated, covering a wide range of fluid properties, pipe diameters, and inclination angles: Eo ∈ [10, 700]; Mo ∈ [1×10–6, 5×103]; ReSL ∈ [–40, 10]; θ ∈ [5°, 90°]. For bubbles in vertical upward flows, the simulated distribution parameter, C0, is successfully compared with an existing model. However, the C0 values of downward and inclined slug flows where the bubble becomes asymmetric are shown to be significantly different from their respective vertical upward flow values, and no current model exists for the fluids simulated here. The main contributions of this work are (1) the relatively large 3D numerical database generated for this type of flow, (2) the study of the asymmetric nature of inclined and some vertical downward slug flows, and (3) the analysis of its impact on the distribution parameter, C0.

Engineering↗
NASA NTRS1995

Visualization of Computational Fluid Dynamics

Scientific visualization serves the dual purpose of exploration and exposition of the results of numerical simulations of fluid flow. Along with the basic visualization process which transforms source data into images, there are four additional components to a complete visualization system: Source Data Processing, User Interface and Control, Presentation, and Information Management. The requirements imposed by the desired mode of operation (i.e. real-time, interactive, or batch) and the source data have their effect on each of these visualization system components. The special requirements imposed by the wide variety and size of the source data provided by the numerical simulation of fluid flow presents an enormous challenge to the visualization system designer. We describe the visualization system components including specific visualization techniques and how the mode of operation and source data requirements effect the construction of computational fluid dynamics visualization systems.

Gerald-Yamasaki, Michael↗
DOE OSTI2025

Subject-specific modeling framework for particle deposition using computational fluid dynamics

Quantifying particle deposition and dose in the respiratory tract requires a physiologically realistic representation and reproducible computational workflows. However, existing modeling frameworks, such as the International Commission on Radiological Protection (ICRP) compartmental models and the Multiple Path Particle Dosimetry (MPPD) tool, lack detailed deposition profiles and subject-specific capabilities. The combination of advances in computer vision algorithms applied to the respiratory tract and Computational Fluid and Particle Dynamics (CFPD) allows high-fidelity simulations of particle behavior in anatomically accurate geometries derived from individual CT scans. The segmentation, preprocessing, and file preparation task for a CFPD simulation was often time-consuming, and no prior studies to-date have yet presented a fully automated framework. This work presents a fully automated workflow to obtain individualized particle deposition profiles in the human respiratory tract. The pipeline starts with segmenting upper and lower airway geometries using morphological and deep learning-based methods, generating three-dimensional (3D) models from CT imaging data. Next, a series of algorithms are presented to quality check and prepare the 3D geometry for a CFD or CFPD simulation. The preprocessing step includes correcting geometric artifacts, enforcing a physically consistent mesh, and automatically identifying and capping multiple outlets, which is required for CFD/CFPD simulations. These processed models are then input into open-source (OpenFOAM) or commercial (StarCCM+) CFD solvers, where flow and transient particle transport equations — including turbulence and particle–wall interactions are solved under realistic breathing conditions. Finally, the resulting particle deposition profiles can be integrated with Monte Carlo radiation transport codes and state-of-the-art computational phantoms to assess organ-specific absorbed doses in scenarios of radioactive aerosol inhalation. The presented work streamlines respiratory tract segmentation, preprocessing for CFD/CFPD simulations, and integration with dose assessment workflows, reducing manual intervention and improving access to high-fidelity, subject-specific modeling. The high precision in predicted particle deposition and dose distributions can improve personalized treatment strategies in respiratory medicine and refine dose estimates for radiation protection.

AI↗
NASA NTRS1991

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I - The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Yee, H. C.↗
NASA NTRS1976

Status and prospects of computational fluid dynamics

The use and limitations on using computational aerodynamics in approximating inviscid linear, inviscid nonlinear, vicous time averaged, and viscous time dependent flow past airfoils, wings, and aircraft is reviewed. The current status of two- and three-dimensional time averaged Navier-Stokes equation is discussed and possible applications for the 1980 and 1985 to 1990 period is projected for three-dimensional applications.

Chapman, D. R.↗
NASA NTRS1985

Status and prospects of computational fluid dynamics for unsteady transonic flow

Applications of computational aerodynamics to aeronautical research, design, and analysis have increased rapidly over the past decade, and these applications offer significant benefits to aeroelasticians. The past developments are traced by means of a number of specific examples, and the trends are projected over the next several years. The crucial factors that limit the present capabilities for unsteady analyses are identified; they include computer speed and memory, algorithm and solution methods, grid generation, turbulence modeling, vortex modeling, data processing, and coupling of the aerodynamic and structural dynamic analyses. The prospects for overcoming these limitations are presented, and many improvements appear to be readily attainable. If so, a complete and reliable numerical simulation of the unsteady, transonic viscous flow around a realistic fighter aircraft configuration could become possible within the next decade. The possibilities of using artificial intelligence concepts to hasten the achievement of this goal are also discussed.

Mccroskey, W. J.↗
NASA NTRS1994

Applications of automatic differentiation in computational fluid dynamics

Automatic differentiation (AD) is a powerful computational method that provides for computing exact sensitivity derivatives (SD) from existing computer programs for multidisciplinary design optimization (MDO) or in sensitivity analysis. A pre-compiler AD tool for FORTRAN programs called ADIFOR has been developed. The ADIFOR tool has been easily and quickly applied by NASA Langley researchers to assess the feasibility and computational impact of AD in MDO with several different FORTRAN programs. These include a state-of-the-art three dimensional multigrid Navier-Stokes flow solver for wings or aircraft configurations in transonic turbulent flow. With ADIFOR the user specifies sets of independent and dependent variables with an existing computer code. ADIFOR then traces the dependency path throughout the code, applies the chain rule to formulate derivative expressions, and generates new code to compute the required SD matrix. The resulting codes have been verified to compute exact non-geometric and geometric SD for a variety of cases. in less time than is required to compute the SD matrix using centered divided differences.

Green, Lawrence L.↗