SEARCH · Engineering Papers

Results for “numerical differentiation”

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.

SourceResearch connectionFrom yearThrough year

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

At least 1,099 records · Page 61

All sources NASA NTRS DOE OSTI

NASA NTRS1991

Calculations of separated 3-D flows with a pressure-staggered Navier-Stokes equations solver

A Navier-Stokes equations solver based on a pressure correction method with a pressure-staggered mesh and calculations of separated three-dimensional flows are presented. It is shown that the velocity pressure decoupling, which occurs when various pressure correction algorithms are used for pressure-staggered meshes, is caused by the ill-conditioned discrete pressure correction equation. The use of a partial differential equation for the incremental pressure eliminates the velocity pressure decoupling mechanism by itself and yields accurate numerical results. Example flows considered are a three-dimensional lid driven cavity flow and a laminar flow through a 90 degree bend square duct. For the lid driven cavity flow, the present numerical results compare more favorably with the measured data than those obtained using a formally third order accurate quadratic upwind interpolation scheme. For the curved duct flow, the present numerical method yields a grid independent solution with a very small number of grid points. The calculated velocity profiles are in good agreement with the measured data.

Kim, S.-W.↗

NASA NTRS1991

Hysteresis and the transition between axisymmetric flow and wave flow in the baroclinic annulus

A numerical model is employed to establish the transitions between axisymmetric flow and wave flow in the rotating, differentially heated annulus experiments of Fein for both rigid lid and free surface cases. It is shown that, for most of the transitions, the method of computing a steady axisymmetric flow and then testing its linear stability to wave disturbance results in good agreement with the experiments. Implications for the investigation of the dynamics of the earth's atmosphere are considered.

Miller, Timothy L.↗

NASA NTRS1991

Calculations of separated 3-D flows with a pressure-staggered Navier-Stokes equations solver

Kim, S.-W.↗

NASA NTRS1993

Nonlinear random response prediction using MSC/NASTRAN

An equivalent linearization technique was incorporated into MSC/NASTRAN to predict the nonlinear random response of structures by means of Direct Matrix Abstract Programming (DMAP) modifications and inclusion of the nonlinear differential stiffness module inside the iteration loop. An iterative process was used to determine the rms displacements. Numerical results obtained for validation on simple plates and beams are in good agreement with existing solutions in both the linear and linearized regions. The versatility of the implementation will enable the analyst to determine the nonlinear random responses for complex structures under combined loads. The thermo-acoustic response of a hexagonal thermal protection system panel is used to highlight some of the features of the program.

Robinson, J. H.↗

NASA NTRS1994

Stability and error estimation for Component Adaptive Grid methods

Component adaptive grid (CAG) methods for solving hyperbolic partial differential equations (PDE's) are discussed in this paper. Applying recent stability results for a class of numerical methods on uniform grids. The convergence of these methods for linear problems on component adaptive grids is established here. Furthermore, the computational error can be estimated on CAG's using the stability results. Using these estimates, the error can be controlled on CAG's. Thus, the solution can be computed efficiently on CAG's within a given error tolerance. Computational results for time dependent linear problems in one and two space dimensions are presented.

Oliger, Joseph↗

NASA NTRS1999

The Thin Oil Film Equation

A thin film of oil on a surface responds primarily to the wall shear stress generated on that surface by a three-dimensional flow. The oil film is also subject to wall pressure gradients, surface tension effects and gravity. The partial differential equation governing the oil film flow is shown to be related to Burgers' equation. Analytical and numerical methods for solving the thin oil film equation are presented. A direct numerical solver is developed where the wall shear stress variation on the surface is known and which solves for the oil film thickness spatial and time variation on the surface. An inverse numerical solver is also developed where the oil film thickness spatial variation over the surface at two discrete times is known and which solves for the wall shear stress variation over the test surface. A One-Time-Level inverse solver is also demonstrated. The inverse numerical solver provides a mathematically rigorous basis for an improved form of a wall shear stress instrument suitable for application to complex three-dimensional flows. To demonstrate the complexity of flows for which these oil film methods are now suitable, extensive examination is accomplished for these analytical and numerical methods as applied to a thin oil film in the vicinity of a three-dimensional saddle of separation.

Brown, James L.↗

NASA NTRS2002

Finite-Strain Fractional-Order Viscoelastic (FOV) Material Models and Numerical Methods for Solving Them

Fraction-order viscoelastic (FOV) material models have been proposed and studied in 1D since the 1930's, and were extended into three dimensions in the 1970's under the assumption of infinitesimal straining. It was not until 1997 that Drozdov introduced the first finite-strain FOV constitutive equations. In our presentation, we shall continue in this tradition by extending the standard, FOV, fluid and solid, material models introduced in 1971 by Caputo and Mainardi into 3D constitutive formula applicable for finite-strain analyses. To achieve this, we generalize both the convected and co-rotational derivatives of tensor fields to fractional order. This is accomplished by defining them first as body tensor fields and then mapping them into space as objective Cartesian tensor fields. Constitutive equations are constructed using both variants for fractional rate, and their responses are contrasted in simple shear. After five years of research and development, we now possess a basic suite of numerical tools necessary to study finite-strain FOV constitutive equations and their iterative refinement into a mature collection of material models. Numerical methods still need to be developed for efficiently solving fraction al-order integrals, derivatives, and differential equations in a finite element setting where such constitutive formulae would need to be solved at each Gauss point in each element of a finite model, which can number into the millions in today's analysis.

Freed, Alan D.↗

NASA NTRS2002

Differential Flatness and Cooperative Tracking in the Lorenz System

In this paper the control of the Lorenz system for both stabilization and tracking problems is studied via feedback linearization and differential flatness. By using the Rayleigh number as the control, only variable physically tunable, a barrier in the controllability of the system is incidentally imposed. This is reflected in the appearance of a singularity in the state transformation. Composite controllers that overcome this difficulty are designed and evaluated. The transition through the manifold defined by such a singularity is achieved by inducing a chaotic response within a boundary layer that contains it. Outside this region, a conventional feedback nonlinear control is applied. In this fashion, the authority of the control is enlarged to the whole. state space and the need for high control efforts is mitigated. In addition, the differential parametrization of the problem is used to track nonlinear functions of one state variable (single tracking) as well as several state variables (cooperative tracking). Control tasks that lead to integrable and non-integrable differential equations for the nominal flat output in steady-state are considered. In particular, a novel numerical strategy to deal with the non-integrable case is proposed. Numerical results validate very well the control design.

Crespo, Luis G.↗

NASA NTRS2019

Accelerated Simulation of Air Pollution Using NVIDIA RAPIDS

Atmospheric chemistry models are a central tool to study and forecast the impact of air pollution on the environment, vegetation, and human health. However, the numerical simulation of chemical kinetics is computationally expensive due to the stiffness of the system of ordinary differential equations that describes atmospheric chemistry. Here we present an alternative approach to the computation of atmospheric chemistry based on machine learning. Our training data set is produced using the NASA Goddard Earth Observing System (GEOS) model with GEOS-Chem chemistry, run on the NASA Center for Climate Simulation (NCCS) Discover supercomputing cluster on 384 Intel Xeon Haswell cores. This model spends more than 50% of total run time on solving atmospheric chemistry. The data set contains as input features the air pollution concentrations before solving the differential equations, together with some key physical parameters such as temperature and sun intensity. As target variables we define the air pollution concentrations after solving the differential equations. Using Dask-cuDF and Dask-XGBoost on the NVIDIA RAPIDS platform on 8 Tesla V100 GPUs, we generate from this training set gradient boosted decision tree models that can reproduce the simulation of chemical kinetics. We do this on the NCCS Advanced Data Analytics Platform (ADAPT) science cloud environment. Our application takes full advantage of recent advances in Dask-XGBoost, such as multi-node and multi-GPU scaling for distributed training with large data sets. The increase in training data size enabled by this is critical to capture the full range of chemical environments encountered across the globe and all annual seasons.The boosted tree models offer good predictability and show many of the features of the full chemistry reference simulation. Further improvements can be achieved through mass balance considerations and by accounting for error correlations. We incorporate the boosted tree models into the GEOS reference model using XGBoost's C API. This enables a seamless integration of the GPU trained models into GEOS-Chem, which is written in Fortran and optimized for use in a massively parallel CPU environment. We show the benefits of this approach and discuss the potential speedup of this machine learning accelerated atmospheric chemistry model.

Keller, Christoph A.↗

NASA NTRS1980

Thermal Pollution Mathematical Model: Verification of One-Dimensional Numerical Model at Lake Keowee - Volume 2

A one dimensional model for studying the thermal dynamics of cooling lakes was developed and verified. The model is essentially a set of partial differential equations which are solved by finite difference methods. The model includes the effects of variation of area with depth, surface heating due to solar radiation absorbed at the upper layer, and internal heating due to the transmission of solar radiation to the sub-surface layers. The exchange of mechanical energy between the lake and the atmosphere is included through the coupling of thermal diffusivity and wind speed. The effects of discharge and intake by power plants are also included. The numerical model was calibrated by applying it to Cayuga Lake. The model was then verified through a long term simulation using Lake Keowee data base. The comparison between measured and predicted vertical temperature profiles for the nine years is good. The physical limnology of Lake Keowee is presented through a set of graphical representations of the measured data base.

Lee, S. S.↗

DOE OSTI2024

Efficient sensitivity analysis of the thermal profile in powder bed fusion of metals using hypercomplex automatic differentiation finite element method

Rapid cyclic temperature fluctuation occurring in powder bed fusion of metals using a laser beam (PBF-LB/M) influences the formation of flaws in printed parts. Consequently, there is a pressing need to enhance the quality of printed parts by developing innovative methodologies that can predict thermal histories and help uncover the intricate relationships between process parameters and thermal profiles. Sensitivity Analysis (SA) emerges as an essential tool for this, offering the potential for process optimization and enhanced quality control. Nonetheless, conventional SA methodologies often incur in excessive computational costs and potential numerical approximation errors. Here, to address this technical challenge, we present a novel method for SA that integrates the HYPercomplex-based Automatic Differentiation (HYPAD) technique with transient thermal simulations conducted via the finite element method (FEM). Leveraging this methodology, we efficiently and accurately perform SA for PBF-LB/M processes in a post-processing step. Compared to traditional methods like Finite Differences (FD), HYPAD-FEM required 96 % less computational time for obtaining sensitivities for 22 process parameters, under a comparative study conducted within the context of the 2018–02 AM benchmark of the National Institute of Standards and Technology. In summary, HYPAD-FEM offers superior efficiency and accuracy in SA over conventional methods, delivering the best sensitivity of a model without the need for step-size selection and problem or parameter-based implementations.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Understanding the distinct impacts of MCS and non-MCS rainfall on the surface water balance in the central US using a numerical water-tagging technique

Warm-season rainfall associated with mesoscale convective systems (MCSs) in the central US is characterized by higher intensity and nocturnal timing compared to rainfall from non-MCS systems, suggesting their potentially different footprints on the land surface. To differentiate the impacts of MCS and non-MCS rainfall on the surface water balance, a water tracer tool embedded in the Noah land surface model with multi-parameterization options (WT-Noah-MP) is used to numerically “tag” water from MCS and non-MCS rainfall separately during April to August (1997-2018) and track their transit in the terrestrial system. From the water-tagging results, over 50% of warm-season rainfall leaves the surface-subsurface system through evapotranspiration by the end of August, but non-MCS rainfall contributes a larger fraction. However, MCS rainfall plays a more important role in generating surface runoff. These differences are mostly attributed to the rainfall intensity differences. The higher intensity MCS rainfall tends to produce more surface runoff through infiltration excess flow and drives a deeper penetration of the rainwater into the soil. Over 70% of the top 10 percentile runoff is contributed by MCS rainfall, demonstrating its important contribution to local flooding. In contrast, lighter intensity non-MCS rainfall resides mostly in the top layer and contributes more to evapotranspiration through soil evaporation. Diurnal timing of rainfall has negligible effects on the flux partitioning for both MCS and non-MCS rainfall. Differences in soil moisture profiles for MCS and non-MCS rainfall and the resultant evapotranspiration suggest differences in their roles in soil moisture-precipitation feedbacks and ecohydrology.

Hu, Huancui↗

NASA NTRS1986

Optimization methods and silicon solar cell numerical models

The goal of this project is the development of an optimization algorithm for use with a solar cell model. It is possible to simultaneously vary design variables such as impurity concentrations, front junction depth, back junctions depth, and cell thickness to maximize the predicted cell efficiency. An optimization algorithm has been developed and interfaced with the Solar Cell Analysis Program in 1 Dimension (SCAPID). SCAPID uses finite difference methods to solve the differential equations which, along with several relations from the physics of semiconductors, describe mathematically the operation of a solar cell. A major obstacle is that the numerical methods used in SCAPID require a significant amount of computer time, and during an optimization the model is called iteratively until the design variables converge to the value associated with the maximum efficiency. This problem has been alleviated by designing an optimization code specifically for use with numerically intensive simulations, to reduce the number of times the efficiency has to be calculated to achieve convergence to the optimal solution. Adapting SCAPID so that it could be called iteratively by the optimization code provided another means of reducing the cpu time required to complete an optimization. Instead of calculating the entire I-V curve, as is usually done in SCAPID, only the efficiency is calculated (maximum power voltage and current) and the solution from previous calculations is used to initiate the next solution.

Girardini, K.↗

NASA NTRS1988

Numerical methods for turbulent flow

It has generally become accepted that the Navier-Strokes equations predict the dynamic behavior of turbulent as well as laminar flows of a fluid at a point in space away form a discontinuity such as a shock wave. Turbulence is also closely related to the phenomena of non-uniqueness of solutions of the Navier-Strokes equations. These second order, nonlinear partial differential equations can be solved analytically for only a few simple flows. Turbulent flow fields are much to complex to lend themselves to these few analytical methods. Numerical methods, therefore, offer the only possibility of achieving a solution of turbulent flow equations. In spite of recent advances in computer technology, the direct solution, by discrete methods, of the Navier-Strokes equations for turbulent flow fields is today, and in the foreseeable future, impossible. Thus the only economically feasible way to solve practical turbulent flow problems numerically is to use statistically averaged equations governing mean-flow quantities. The objective is to study some recent developments relating to the use of numerical methods to study turbulent flow.

Turner, James C., Jr.↗

DOE OSTI2021

Hierarchical model reduction driven by a proper orthogonal decomposition for parametrized advection-diffusion-reaction problems

This work combines the Hierarchical Model (HiMod) reduction technique with a standard Proper Orthogonal Decomposition (POD) to solve parametrized partial differential equations for the modeling of advection-diffusion-reaction phenomena in elongated domains (e.g., pipes). This combination leads to what we define as HiPOD model reduction, which merges the reliability of HiMod reduction with the computational efficiency of POD. Two HiPOD techniques are presented and assessed by an extensive numerical verification.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1972

A discrete variable approximation to minimum weight panel designs subject to a supersonic flutter-speed constraint.

A numerical technique is presented that determines the minimum-weight thickness distribution for a one-dimensional, simply supported panel subjected to a parallel supersonic flow on one side. An aeroelastic eigenvalue, which characterizes flutter speed, is held constant. The governing differential equations are approximated by sets of difference equations adjoined to the weight function via a penalty function. A conjugate gradient method then solves the resulting sequence of unconstrained minimization problems. Numerical results are obtained without imposing a minimum thickness constraint; the effect of constant inplane tensile and compressive stresses is also investigated.

Pierson, B. L.↗

NASA NTRS2001

Candle Flames in Microgravity

The goal of this work is to study both experimentally and numerically the behavior of a candle flame burning in a microgravity environment. Two space experiments (Shuttle and Mir) have shown the candle flame in microgravity to be small (approximately 1.5 cm diameter), dim blue, and hemispherical. Near steady flames with very long flame lifetimes (up to 45 minutes in some tests) existed for many of the tests. Most of the flames spontaneously oscillated with a period of approximately 1 Hz just prior to extinction). In a previous model of candle flame in microgravity, a porous sphere wetted with liquid fuel simulated the evaporating wick. The sphere, with a temperature equal to the boiling temperature of the fuel, was at the end of an inert cone that had a prescribed temperature. This inert cone produces the quenching effect of the candle wax in the real configuration. Although the computed flame shape resembled that observed in the microgravity experiment, the model was not able to differentiate the effect of wick geometry, e.g., a long vs. a short wick. This paper presents recent developments in the numerical model of the candle flame. The primary focus has been to more realistically account for the actual shape of the candle.

Dietrich, D. L.↗

NASA NTRS1972

Earth zonal harmonics from rapid numerical analysis of long satellite arcs

A zonal geopotential is presented to degree 21 from evaluation of mean elements for 21 satellites including 2 of low inclination. Each satellite is represented by an arc of at least one apsidal rotation. The lengths range from 200 to 800 days. Differential correction of the initial elements in all of the arcs, together with radiation pressure and atmospheric drag coefficients, was accomplished simultaneously with the correction for the zonal harmonics. The satellite orbits and their variations are generated by numerical integration of the Lagrange equations for mean elements. Disturbances due to precession and nutation of the earth's pole, atmospheric drag, radiation pressure and luni-solar gravity are added at from 1- to 8-day intervals in the integrated orbits. The results agree well with recent solutions from other authors using different methods and different satellite sets.

Wagner, C. A.↗