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NASA NTRSDate not supplied

Discretization Error Estimation and Control for Farfield Acoustic Signatures

We investigate the utility of adjoint-based error estimates for sonic boom farfield simulations governed by solutions of the augmented Burgers’ equation. Solution of this nonlinear system uses operator splitting with a second-order finite volume discretization in space and second-order Runge-Kutta time marching, while the absorption and molecular relaxation are solved using second-order central differencing. The discretization error in selected ground sonic boom cost functionals is estimated using the method of adjoint-weighted residuals. Key elements of the implementation process are emphasized with details provided on the practical aspects as appliedto the sonic boom farfield propagation. We establish the accuracy of the adjoint solutions usingcomplex step and finite difference approaches, and examine the accuracy of the error estimates using analytical N-wave solutions. We then apply it to a pressure waveform corresponding to the X-59 research aircraft. The investigations demonstrate that the method of adjoint-weighted residuals accurately predicts the level of discretization error present in sonic boom farfield simulations while offering insight into which features of the near field signal are the primary drivers of ground noise metrics. The numerical results indicate that at sampling frequencies as low as50kHz, discretization error in the propagation is under 0.01 dB[A] for realistically complex examples.

CST↗
NASA NTRS1998

A Novel Four-Node Quadrilateral Smoothing Element for Stress Enhancement and Error Estimation

A four-node, quadrilateral smoothing element is developed based upon a penalized-discrete-least-squares variational formulation. The smoothing methodology recovers C1-continuous stresses, thus enabling effective a posteriori error estimation and automatic adaptive mesh refinement. The element formulation is originated with a five-node macro-element configuration consisting of four triangular anisoparametric smoothing elements in a cross-diagonal pattern. This element pattern enables a convenient closed-form solution for the degrees of freedom of the interior node, resulting from enforcing explicitly a set of natural edge-wise penalty constraints. The degree-of-freedom reduction scheme leads to a very efficient formulation of a four-node quadrilateral smoothing element without any compromise in robustness and accuracy of the smoothing analysis. The application examples include stress recovery and error estimation in adaptive mesh refinement solutions for an elasticity problem and an aerospace structural component.

Tessler, A.↗
NASA NTRS2011

Error Estimate of the Ares I Vehicle Longitudinal Aerodynamic Characteristics Based on Turbulent Navier-Stokes Analysis

Numerical predictions of the longitudinal aerodynamic characteristics for the Ares I class of vehicles, along with the associated error estimate derived from an iterative convergence grid refinement, are presented. Computational results are based on the unstructured grid, Reynolds-averaged Navier-Stokes flow solver USM3D, with an assumption that the flow is fully turbulent over the entire vehicle. This effort was designed to complement the prior computational activities conducted over the past five years in support of the Ares I Project with the emphasis on the vehicle s last design cycle designated as the A106 configuration. Due to a lack of flight data for this particular design s outer mold line, the initial vehicle s aerodynamic predictions and the associated error estimates were first assessed and validated against the available experimental data at representative wind tunnel flow conditions pertinent to the ascent phase of the trajectory without including any propulsion effects. Subsequently, the established procedures were then applied to obtain the longitudinal aerodynamic predictions at the selected flight flow conditions. Sample computed results and the correlations with the experimental measurements are presented. In addition, the present analysis includes the relevant data to highlight the balance between the prediction accuracy against the grid size and, thus, the corresponding computer resource requirements for the computations at both wind tunnel and flight flow conditions. NOTE: Some details have been removed from selected plots and figures in compliance with the sensitive but unclassified (SBU) restrictions. However, the content still conveys the merits of the technical approach and the relevant results.

Aerodynamic characteristics↗
NASA NTRS2006

Improved Soundings and Error Estimates using AIRS/AMSU Data

AIRS was launched on EOS Aqua on May 4, 2002, together with AMSU A and HSB, to form a next generation polar orbiting infrared and microwave atmospheric sounding system. The primary products of AIRS/AMSU are twice daily global fields of atmospheric temperature-humidity profiles, ozone profiles, sea/land surface skin temperature, and cloud related parameters including OLR. The sounding goals of AIRS are to produce 1 km tropospheric layer mean temperatures with an rms error of 1 K, and layer precipitable water with an rms error of 20 percent, in cases with up to 80 percent effective cloud cover. The basic theory used to analyze AIRS/AMSU/HSB data in the presence of clouds, called the at-launch algorithm, and a post-launch algorithm which differed only in the minor details from the at-launch algorithm, have been described previously. The post-launch algorithm, referred to as AIRS Version 4.0, has been used by the Goddard DAAC to analyze and distribute AIRS retrieval products. In this paper we show progress made toward the AIRS Version 5.0 algorithm which will be used by the Goddard DAAC starting late in 2006. A new methodology has been developed to provide accurate case by case error estimates for retrieved geophysical parameters and for the channel by channel cloud cleared radiances used to derive the geophysical parameters from the AIRS/AMSU observations. These error estimates are in turn used for quality control of the derived geophysical parameters and clear column radiances. Improvements made to the retrieval algorithm since Version 4.0 are described as well as results comparing Version 5.0 retrieval accuracy and spatial coverage with those obtained using Version 4.0.

Susskind, Joel↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal estimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed.

Rodriguez, G.↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal esimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed. While this paper focuses on theoretical developments, applications arising in the area of large structure static shape determination are contained in a closely related paper (Rodriguez and Scheid, 1982).

Rodriguez, G.↗
NASA NTRS2007

Improved Atmospheric Soundings and Error Estimates from Analysis of AIRS/AMSU Data

The AIRS Science Team Version 5.0 retrieval algorithm became operational at the Goddard DAAC in July 2007 generating near real-time products from analysis of AIRS/AMSU sounding data. This algorithm contains many significant theoretical advances over the AIRS Science Team Version 4.0 retrieval algorithm used previously. Three very significant developments of Version 5 are: 1) the development and implementation of an improved Radiative Transfer Algorithm (RTA) which allows for accurate treatment of non-Local Thermodynamic Equilibrium (non-LTE) effects on shortwave sounding channels; 2) the development of methodology to obtain very accurate case by case product error estimates which are in turn used for quality control; and 3) development of an accurate AIRS only cloud clearing and retrieval system. These theoretical improvements taken together enabled a new methodology to be developed which further improves soundings in partially cloudy conditions, without the need for microwave observations in the cloud clearing step as has been done previously. In this methodology, longwave C02 channel observations in the spectral region 700 cm-' to 750 cm-' are used exclusively for cloud clearing purposes, while shortwave C02 channels in the spectral region 2195 cm-' to 2395 cm-' are used for temperature sounding purposes. The new methodology for improved error estimates and their use in quality control is described briefly and results are shown indicative of their accuracy. Results are also shown of forecast impact experiments assimilating AIRS Version 5.0 retrieval products in the Goddard GEOS 5 Data Assimilation System using different quality control thresholds.

Susskind, Joel↗
NASA NTRS1999

A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

Larson, Mats G.↗
NASA NTRS1979

State and model error estimation for distributed parameter systems

In-flight estimation of large structure model errors in order to detect inevitable deficiencies in large structure controller/estimator models is discussed. Such an estimation process is particularly applicable in the area of shape control system design required to maintain a prescribed static structural shape and, in addition, suppress dynamic disturbances due to the vehicle vibrational modes. The paper outlines a solution to the problem of static shape estimation where the vehicle shape must be reconstructed from a set of measurements discretely located throughout the structure. The estimation process is based on the principle of least-squares that inherently contains the definition and explicit computation of model error estimates that are optimal in some sense. Consequently, a solution is provided for the problem of estimation of static model errors (e.g., external loads). A generalized formulation applicable to distributed parameters systems is first worked out and then applied to a one-dimensional beam-like structural configuration.

Rodriguez, G.↗
NASA NTRS1994

Error estimation and adaptive mesh refinement for parallel analysis of shell structures

The formulation and application of element-level, element-independent error indicators is investigated. This research culminates in the development of an error indicator formulation which is derived based on the projection of element deformation onto the intrinsic element displacement modes. The qualifier 'element-level' means that no information from adjacent elements is used for error estimation. This property is ideally suited for obtaining error values and driving adaptive mesh refinements on parallel computers where access to neighboring elements residing on different processors may incur significant overhead. In addition such estimators are insensitive to the presence of physical interfaces and junctures. An error indicator qualifies as 'element-independent' when only visible quantities such as element stiffness and nodal displacements are used to quantify error. Error evaluation at the element level and element independence for the error indicator are highly desired properties for computing error in production-level finite element codes. Four element-level error indicators have been constructed. Two of the indicators are based on variational formulation of the element stiffness and are element-dependent. Their derivations are retained for developmental purposes. The second two indicators mimic and exceed the first two in performance but require no special formulation of the element stiffness mesh refinement which we demonstrate for two dimensional plane stress problems. The parallelizing of substructures and adaptive mesh refinement is discussed and the final error indicator using two-dimensional plane-stress and three-dimensional shell problems is demonstrated.

Keating, Scott C.↗
NASA NTRS1993

A priori error estimates for an hp-version of the discontinuous Galerkin method for hyperbolic conservation laws

A priori error estimates are derived for hp-versions of the finite element method for discontinuous Galerkin approximations of a model class of linear, scalar, first-order hyperbolic conservation laws. These estimates are derived in a mesh dependent norm in which the coefficients depend upon both the local mesh size h(sub K) and a number p(sub k) which can be identified with the spectral order of the local approximations over each element.

Bey, Kim S.↗
NASA NTRS1999

Adaptive Error Estimation in Linearized Ocean General Circulation Models

Data assimilation methods are routinely used in oceanography. The statistics of the model and measurement errors need to be specified a priori. This study addresses the problem of estimating model and measurement error statistics from observations. We start by testing innovation based methods of adaptive error estimation with low-dimensional models in the North Pacific (5-60 deg N, 132-252 deg E) to TOPEX/POSEIDON (TIP) sea level anomaly data, acoustic tomography data from the ATOC project, and the MIT General Circulation Model (GCM). A reduced state linear model that describes large scale internal (baroclinic) error dynamics is used. The methods are shown to be sensitive to the initial guess for the error statistics and the type of observations. A new off-line approach is developed, the covariance matching approach (CMA), where covariance matrices of model-data residuals are "matched" to their theoretical expectations using familiar least squares methods. This method uses observations directly instead of the innovations sequence and is shown to be related to the MT method and the method of Fu et al. (1993). Twin experiments using the same linearized MIT GCM suggest that altimetric data are ill-suited to the estimation of internal GCM errors, but that such estimates can in theory be obtained using acoustic data. The CMA is then applied to T/P sea level anomaly data and a linearization of a global GFDL GCM which uses two vertical modes. We show that the CMA method can be used with a global model and a global data set, and that the estimates of the error statistics are robust. We show that the fraction of the GCM-T/P residual variance explained by the model error is larger than that derived in Fukumori et al.(1999) with the method of Fu et al.(1993). Most of the model error is explained by the barotropic mode. However, we find that impact of the change in the error statistics on the data assimilation estimates is very small. This is explained by the large representation error, i.e. the dominance of the mesoscale eddies in the T/P signal, which are not part of the 21 by 1" GCM. Therefore, the impact of the observations on the assimilation is very small even after the adjustment of the error statistics. This work demonstrates that simult&neous estimation of the model and measurement error statistics for data assimilation with global ocean data sets and linearized GCMs is possible. However, the error covariance estimation problem is in general highly underdetermined, much more so than the state estimation problem. In other words there exist a very large number of statistical models that can be made consistent with the available data. Therefore, methods for obtaining quantitative error estimates, powerful though they may be, cannot replace physical insight. Used in the right context, as a tool for guiding the choice of a small number of model error parameters, covariance matching can be a useful addition to the repertory of tools available to oceanographers.

Chechelnitsky, Michael Y.↗