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NASA NTRS2014

Entropy Stable Wall Boundary Conditions for the Compressible Navier-Stokes Equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite volume, finite difference, discontinuous Galerkin, and flux reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.

Parsani, Matteo↗

NASA NTRS2015

Entropy Stable Wall Boundary Conditions for the Three-Dimensional Compressible Navier-Stokes Equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.

Parsani, Matteo↗

NASA NTRS1991

Development of a CFD Code for Analysis of Fluid Dynamic Forces in Seals

The aim is to develop a 3-D computational fluid dynamics (CFD) code for the analysis of fluid flow in cylindrical seals and evaluation of the dynamic forces on the seals. This code is expected to serve as a scientific tool for detailed flow analysis as well as a check for the accuracy of the 2D industrial codes. The features necessary in the CFD code are outlined. The initial focus was to develop or modify and implement new techniques and physical models. These include collocated grid formulation, rotating coordinate frames and moving grid formulation. Other advanced numerical techniques include higher order spatial and temporal differencing and an efficient linear equation solver. These techniques were implemented in a 2D flow solver for initial testing. Several benchmark test cases were computed using the 2D code, and the results of these were compared to analytical solutions or experimental data to check the accuracy. Tests presented here include planar wedge flow, flow due to an enclosed rotor, and flow in a 2D seal with a whirling rotor. Comparisons between numerical and experimental results for an annular seal and a 7-cavity labyrinth seal are also included.

Athavale, Mahesh M.↗

NASA NTRS1994

Validation of a technique for estimating outgoing longwave radiation from HIRS radiance observations

Simultaneous observations by the Earth Radiation Budget Experiment (ERBE) scanning radiometer and the High-Resolution Infrared Sounder (HIRS) on board the NOAA-9 spacecraft have been used to validate a multispectral technique for estimating the outgoing longwave radiation (OLR) from the earth-atmosphere system. Results from approximately 100 000 collocated observations show that the HIRS technique provides instantaneous OLR estimates that agree with the ERBE observations just as well as different ERBE scanners agree with each other--about 5 W m(exp -2) rms. Although there are differences between the HIRS and ERBE estimates that depend upon the scene type and time of day, the HIRS technique explained more than 99% of the variance of the ERBE observations for both day and night observations. The results suggest that the HIRS OLR technique is a suitable replacement for the Advanced Very High Resolution Radiometer technique now used by the National Oceanic and Atmospheric Administration for operational estimates of the OLR.

Ellingson, Robert G.↗

NASA NTRS1986

A spectral multi-domain technique with application to generalized curvilinear coordinates

Spectral collocation methods have proven to be efficient discretization schemes for many aerodynamic and fluid mechanic problems. The high order accuracy and resolution shown by these methods allows one to obtain engineering accuracy solutions on coarse meshes, or alternatively, to obtain solutions with very small error. One drawback to these techniques was the requirement that a complicated physical domain must map into a simple computational domain for discretization. This mapping must be smooth if the high order accuracy and expontential convergence rates associated with spectral methods are to be preserved. Additionally even smooth stretching transformations can decrease the accuracy of a spectral method, if the stretching is severe. A further difficulty with spectral methods was in their implementation on parallel processing computers, where efficient spectral algorithms were lacking. The above restrictions are overcome by splitting the domain into regions, each of which preserve the advantages of spectral collocation, and allow the ratio of the mesh spacing between regions to be several orders of magnitude higher than allowable in a single domain. Such stretchings would be required to resolve the thin viscous region in an external aerodynamic problem. Adjoining regions are interfaced by enforcing a global flux balance which preserves high-order continuity of the solution, regardless of the type of the equations being solved.

Macaraeg, M. G.↗

NASA NTRS1985

The stress distribution in pin-loaded orthotropic plates

The performance of mechanically fastened composite joints was studied. Specially, a single-bolt connector was modeled as a pin-loaded, infinite plate. The model that was developed used two dimensional, complex variable, elasticity techniques combined with a boundary collocation procedure to produce solutions for the problem. Through iteration, the boundary conditions were satisfied and the stresses in the plate were calculated. Several graphite epoxy laminates were studied. In addition, parameters such as the pin modulus, coefficient of friction, and pin-plate clearance were varied. Conclusions drawn from this study indicate: (1) the material properties (i.e., laminate configuration) of the plate alter the stress state and, for highly orthotropic materials, the contact stress deviates greatly from the cosinusoidal distribution often assumed; (2) friction plays a major role in the distribution of stresses in the plate; (3) reversing the load direction also greatly effects the stress distribution in the plate; (4) clearance (or interference) fits change the contact angle and thus the location of the peak hoop stress; and (5) a rigid pin appears to be a good assumption for typical material systems.

Klang, E. C.↗

DOE OSTI2013

A GENERALIZED STOCHASTIC COLLOCATION APPROACH TO CONSTRAINED OPTIMIZATION FOR RANDOM DATA IDENTIFICATION PROBLEMS

We present a scalable, parallel mechanism for stochastic identification/control for problems constrained by partial differential equations with random input data. Several identification objectives will be discussed that either minimize the expectation of a tracking cost functional or minimize the difference of desired statistical quantities in the appropriate $L^p$ norm, and the distributed parameters/control can both deterministic or stochastic. Given an objective we prove the existence of an optimal solution, establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations. The modeling process may describe the solution in terms of high dimensional spaces, particularly in the case when the input data (coefficients, forcing terms, boundary conditions, geometry, etc) are affected by a large amount of uncertainty. For higher accuracy, the computer simulation must increase the number of random variables (dimensions), and expend more effort approximating the quantity of interest in each individual dimension. Hence, we introduce a novel stochastic parameter identification algorithm that integrates an adjoint-based deterministic algorithm with the sparse grid stochastic collocation FEM approach. This allows for decoupled, moderately high dimensional, parameterized computations of the stochastic optimality system, where at each collocation point, deterministic analysis and techniques can be utilized. The advantage of our approach is that it allows for the optimal identification of statistical moments (mean value, variance, covariance, etc.) or even the whole probability distribution of the input random fields, given the probability distribution of some responses of the system (quantities of physical interest). Our rigorously derived error estimates, for the fully discrete problems, will be described and used to compare the efficiency of the method with several other techniques. Numerical examples illustrate the theoretical results and demonstrate the distinctions between the various stochastic identification objectives.

Webster, Clayton G↗

DOE OSTI2012

An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Stochastic Partial Differential Equations

Accurate predictive simulations of complex real world applications require numerical approximations to first, oppose the curse of dimensionality and second, converge quickly in the presence of steep gradients, sharp transitions, bifurcations or finite discontinuities in high-dimensional parameter spaces. In this paper we present a novel multi-dimensional multi-resolution adaptive (MdMrA) sparse grid stochastic collocation method, that utilizes hierarchical multiscale piecewise Riesz basis functions constructed from interpolating wavelets. The basis for our non-intrusive method forms a stable multiscale splitting and thus, optimal adaptation is achieved. Error estimates and numerical examples will used to compare the efficiency of the method with several other techniques.

adaptive sparse grids↗

NASA NTRS1989

Regional quasigeoid determination in Northern Germany and comparison with GPS

For the northern part of the Federal Republic of Germany, new quasigeoid solutions were computed by least squares collocation and fast Fourier transformation (FFT) techniques using point and mean gravity data, a digital terrain model, and a global geopotential model. As severe accuracy limitations for precise regional quasigeoid determination come from global model uncertainties, different geopotential models were investigated by combining them with gravimetric data and comparing the quasigeoid heights with GPS and leveling. Optimum results were obtained by a global model tailored to gravity data in Europe. Collocation and FFT results based on this model agree well. The comparison with GPS and leveling yields rms discrepancies of + or - 2 cm over approximately 400 km range.

Denker, Heiner↗

DOE OSTI2019

Retrieval of warm precipitation microphysics

Intrieri et al. (1993) and later O’Connor et al. (2005) proposed a technique to constrain water drop size distribution using lidar backscatter (related to water drop cross-section) and Doppler spectral width (related to the width of the water drop size distribution). This radar-lidar technique can be used to estimate precipitation rate and microphysical characteristics at all levels in the subcloud layer when collocated radar and ceilometer observations are available. We apply this technique to the vertically pointing ARM ceilometer lidar and Ka-band Zenith Radar (KAZR2) pair. The O’Connor et al. (2005) technique requires ceilometer backscatter to be calibrated and remapped to the radar spatio-temporal resolution (here 2 s x 30 m). Ceilometer backscatter is calibrated following a variation of the O'Connor et al. (2004) technique by scaling observed path-integrated backscatter in thick stratocumulus to match theoretical cloud lidar ratio values. Satisfactory conditions for ceilometer backscatter calibration are identified as the first (in time) 20-min periods each day with standard deviation of lidar ratio smaller than 1.5. The observed backscatter during the “satisfactory 20-min period” are input to Hogan (2006)’s multi scattered model to determine a daily backscatter calibration factor. For days where satisfactory conditions are not observed, a climatological calibration factor of 1.35 is used to calibrate the observed backscatter. Calibrated ceilometer backscatter is subsequently mapped on the KAZR2 time-height grid using a nearest-neighbor approach. Following Kollias et al. (2019), KAZR2 calibration is performed using collocated surface-based Parsivel laser disdrometer-equivalent radar reflectivity estimates during light precipitation events. This radar-lidar technique generates time-height maps of precipitation rate from 200 m above ground level to 90 m below cloud base height that are filtered for aerosol contamination.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1988

Parameter estimation for boundary value problems by integral equations of the second kind

This paper is concerned with the parameter estimation for boundary integral equations of the second kind. The parameter estimation technique through use of the spline collocation method is proposed. Based on the compactness assumption imposed on the parameter space, the convergence analysis for the numerical method of parameter estimation is discussed. The results obtained here are applied to a boundary parameter estimation for 2-D elliptic systems.

Kojima, Fumio↗

NASA NTRS2004

Scalable collaborative risk management technology for complex critical systems

We describe here our project and plans to develop methods, software tools, and infrastructure tools to address challenges relating to geographically distributed software development. Specifically, this work is creating an infrastructure that supports applications working over distributed geographical and organizational domains and is using this infrastructure to develop a tool that supports project development using risk management and analysis techniques where the participants are not collocated.

requirements prioritization↗

DOE OSTI2020

A Hybrid Lagrangian/Eulerian Collocated Velocity Advection and Projection Method for Fluid Simulation

Abstract We present a hybrid particle/grid approach for simulating incompressible fluids on collocated velocity grids. Our approach supports both particle‐based Lagrangian advection in very detailed regions of the flow and efficient Eulerian grid‐based advection in other regions of the flow. A novel Backward Semi‐Lagrangian method is derived to improve accuracy of grid based advection. Our approach utilizes the implicit formula associated with solutions of the inviscid Burgers’ equation. We solve this equation using Newton's method enabled by C 1 continuous grid interpolation. We enforce incompressibility over collocated, rather than staggered grids. Our projection technique is variational and designed for B‐spline interpolation over regular grids where multiquadratic interpolation is used for velocity and multilinear interpolation for pressure. Despite our use of regular grids, we extend the variational technique to allow for cut‐cell definition of irregular flow domains for both Dirichlet and free surface boundary conditions.

Gagniere, S.↗

NASA NTRS1983

Spectral methods for partial differential equations

Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized.

Hussaini, M. Y.↗

NASA NTRS1984

Spectral methods for partial differential equations

Hussaini, M. Y.↗

NASA NTRS1985

Recent applications of spectral methods in fluid dynamics

Origins of spectral methods, especially their relation to the method of weighted residuals, are surveyed. Basic Fourier and Chebyshev spectral concepts are reviewed and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic and mixzed type. Fluid dynamical applications are emphasized.

Zang, T. A.↗

NASA NTRS1984

Spectral methods for the Euler equations: Fourier methods and shock-capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one dimensional, periodic astrophysical ""nozzle'' problem.

Hussaini, M. Y.↗

NASA NTRS1990

Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming

A new method is described for the determination of optimal spacecraft trajectories in an inverse-square field using finite, fixed thrust. The method employs a recently developed optimization technique which uses a piecewise polynomial representation for the state and controls, and collocation, thus converting the optimal control problem into a nonlinear programming problem, which is solved numerically. This technique has been modified to provide efficient handling of those portions of the trajectory which can be determined analytically, i.e., the coast arcs. Among the problems that have been solved using this method are optimal rendezvous and transfer (including multirevolution cases) and optimal multiburn orbit insertion from hyperbolic approach.

Enright, Paul J.↗