Search NASA⌕ Search

SEARCH · Search NASA

Results for “Discretization error”

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 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS2010

Exploring Discretization Error in Simulation-Based Aerodynamic Databases

This work examines the level of discretization error in simulation-based aerodynamic databases and introduces strategies for error control. Simulations are performed using a parallel, multi-level Euler solver on embedded-boundary Cartesian meshes. Discretization errors in user-selected outputs are estimated using the method of adjoint-weighted residuals and we use adaptive mesh refinement to reduce these errors to specified tolerances. Using this framework, we examine the behavior of discretization error throughout a token database computed for a NACA 0012 airfoil consisting of 120 cases. We compare the cost and accuracy of two approaches for aerodynamic database generation. In the first approach, mesh adaptation is used to compute all cases in the database to a prescribed level of accuracy. The second approach conducts all simulations using the same computational mesh without adaptation. We quantitatively assess the error landscape and computational costs in both databases. This investigation highlights sensitivities of the database under a variety of conditions. The presence of transonic shocks or the stiffness in the governing equations near the incompressible limit are shown to dramatically increase discretization error requiring additional mesh resolution to control. Results show that such pathologies lead to error levels that vary by over factor of 40 when using a fixed mesh throughout the database. Alternatively, controlling this sensitivity through mesh adaptation leads to mesh sizes which span two orders of magnitude. We propose strategies to minimize simulation cost in sensitive regions and discuss the role of error-estimation in database quality.

Aftosmis, Michael J.↗
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 NTRS1995

Analysis of discretization errors in LES

All numerical simulations of turbulence (DNS or LES) involve some discretization errors. The integrity of such simulations therefore depend on our ability to quantify and control such errors. In the classical literature on analysis of errors in partial differential equations, one typically studies simple linear equations (such as the wave equation or Laplace's equation). The qualitative insight gained from studying such simple situations is then used to design numerical methods for more complex problems such as the Navier-Stokes equations. Though such an approach may seem reasonable as a first approximation, it should be recognized that strongly nonlinear problems, such as turbulence, have a feature that is absent in linear problems. This feature is the simultaneous presence of a continuum of space and time scales. Thus, in an analysis of errors in the one dimensional wave equation, one may, without loss of generality, rescale the equations so that the dependent variable is always of order unity. This is not possible in the turbulence problem since the amplitudes of the Fourier modes of the velocity field have a continuous distribution. The objective of the present research is to provide some quantitative measures of numerical errors in such situations. Though the focus of this work is LES, the methods introduced here can be just as easily applied to DNS. Errors due to discretization of the time-variable are neglected for the purpose of this analysis.

Ghosal, Sandip↗
NASA NTRSDate not supplied

Goal-Oriented Discretization Error Control in Coupled Nearfield-Farfield Low-Boom Simulations

The method of adjoint weighted residuals is used to determine the level of discretization error in loudness predictions of sonic booms on the ground. We analyze the standard nearfied-farfield domain decomposition approach. In the nearfield domain, the three-dimensional Euler equations are solved to obtain a pressure signature generated by the aircraft. In the farfield, this waveform is propagated through the atmosphere to the ground by solving the augmented Burgers’ equation. Loudness is characterized using weighted sound-exposure-level metrics. We formulate discretization error estimates for the ground signature and loudness metrics for this one-way coupled system. Although the nearfield solution is independent of the farfield, the adjoint formulation for the coupled system provides feedback from the farfield to identify high-error regions in the nearfield. The results demonstrate that the discrete adjoint implementation is asymptotically consistent and provides reliable error estimates. Furthermore, we show how the error can be controlled through adaptive refinement of the nearfield mesh. The approach is evaluated on two- and three-dimensional problems, including the X-59 flight demonstration aircraft.

CST↗
NASA NTRS1977

Differential altimetry for satellite orbit determination

Differential altimetry is concerned with the employment of differenced satellite altimeter measurements at orbit ground trace intersections. The employment of this procedure makes it possible to eliminate two of the major error sources found in direct altimetry. Previous applications have not included the appropriate dynamic constraints required to account for correlations due to satellite orbit motion. A description is given of an investigation in which these correlations are included. The methodology produced is consistent with the dynamic environment. The regional or local limitations of previous approaches are overcome by extending the technique to the global scale. Attention is given to the description of the data type, the geometric topography height, altimeter errors, discretization errors, an approximate orbit determination problem, and a comparison of differenced altimeter measurements for retrograde and prograde orbits.

Hagar, H., Jr.↗
NASA NTRS2010

Accuracy Analysis for Finite-Volume Discretization Schemes on Irregular Grids

A new computational analysis tool, downscaling test, is introduced and applied for studying the convergence rates of truncation and discretization errors of nite-volume discretization schemes on general irregular (e.g., unstructured) grids. The study shows that the design-order convergence of discretization errors can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all. The downscaling test is a general, efficient, accurate, and practical tool, enabling straightforward extension of verification and validation to general unstructured grid formulations. It also allows separate analysis of the interior, boundaries, and singularities that could be useful even in structured-grid settings. There are several new findings arising from the use of the downscaling test analysis. It is shown that the discretization accuracy of a common node-centered nite-volume scheme, known to be second-order accurate for inviscid equations on triangular grids, degenerates to first order for mixed grids. Alternative node-centered schemes are presented and demonstrated to provide second and third order accuracies on general mixed grids. The local accuracy deterioration at intersections of tangency and in flow/outflow boundaries is demonstrated using the DS tests tailored to examining the local behavior of the boundary conditions. The discretization-error order reduction within inviscid stagnation regions is demonstrated. The accuracy deterioration is local, affecting mainly the velocity components, but applies to any order scheme.

Diskin, Boris↗
NASA NTRS1989

Control of the errors of discretization and idealization in finite element analysis

Understanding of the basic principles which control errors of discretization in finite element analysis has increased very substantially since 1980. The main milestones were: (1) development of the theoretical basis of p-extensions (1981); (2) understanding of the proper interplay between mesh design and assignment of polynomial degree to elements. Practical realization of exponential convergence rates, independently of the smoothness of the exact solution (1984); and (3) industrial experience with the new finite element technology known as the p- or hp-version of the finite element method: General Dynamics reported thirty- to forty-fold savings in terms of human time and large savings in computer time (1986). Lockheed reported favorably on their evaluation of error estimation and quality control capabilities of the p-version in industrial settings (1987). The gains in our understanding of how to control the errors of discretization represent only half of the control necessary to ensure that a numerical model is in fact an accurate representation of the corresponding physical system. Control of the errors of idealization is equally important. A brief overview of the main ideas of how to ensure the quality and reliability of mathematical models of structural systems is presented.

Szabo, Barna A.↗
NASA NTRS2010

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. The grids range from regular grids to irregular grids, including mixed-element grids and grids with random perturbations of nodes. Accuracy, complexity, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes with weighted and unweighted least-squares (LSQ) methods for gradient reconstruction and six cell-centered schemes two node-averaging with and without clipping and four schemes that employ different stencils for LSQ gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the LSQ stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered (NC) schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The NC schemes converge with second order on regular and/or triangular grids and with first order on perturbed quadrilaterals and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and secondorder discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors converge with second order and are small for the CC-NN, CC-SA, and CC-FA schemes on all grids and for NC schemes on triangular grids; the discretization errors of the CC-NA scheme without clipping do not converge on irregular grids. Accurate gradient reconstruction can be achieved by introducing a local approximate mapping; without approximate mapping, only the NC scheme with weighted LSQ method provides accurate gradients. Defect correction iterations for the CC-NA scheme without clipping diverge; for the NC scheme with weighted LSQ method, the iterations either diverge or converge very slowly. The best option in curved geometries is the CC-SA scheme that offers low complexity, second-order discretization errors, and fast convergence.

Diskin, Boris↗
NASA NTRSDate not supplied

Comparing Unstructured Adaptive Mesh Solutions for the High Lift Common Research Model Airfoil

Discretization error is a common source of uncertainty in Computational Fluid Dynamics (CFD) analyses. Traditional means of controlling discretization error through fixed-mesh refinement studies has proven to be difficult particularly when modeling complex geometries and flow fields. One reason for this is that mesh generation in today’s production CFD workflow is often a labor intensive process that is heavily dependent on user judgment. Unstructured mesh adaptation is known to be an efficient way to control discretization errors in CFD. Adaptive methods replace user based decision making with automated processes that optimize a mesh to reduce discretization error. This paper compares the application of multiple solution adaptive techniques in combination with multiple flow solvers to solve for the flow field about a 2D airfoil section of the NASA High-Lift Common Research Model (HL-CRM). By driving the adaptive mesh processes to a similar level of mesh convergence, the ability to achieve consistent results between multiple adaptive techniques and flow solvers is demonstrated. Mesh convergence for the various adaptive mesh approaches is compared identifying potential areas for improvement and providing mesh generation guidance for future workshops.

mesh adaptation high-lift 2D airfoil↗
NASA NTRSDate not supplied

Comparing Unstructured Adaptive Mesh Solutions for the High Lift Common Research Model Airfoil

Discretization error is a common source of uncertainty in Computational Fluid Dynamics (CFD) analyses. Traditional means of controlling discretization error through fixed-mesh refinement studies has proven to be difficult particularly when modeling complex geometries and flow fields. One reason for this is that mesh generation in today’s production CFD workflow is often a labor intensive process that is heavily dependent on user judgment. Unstructured mesh adaptation is known to be an efficient way to control discretization errors in CFD. Adaptive methods replace user based decision making with automated processes that optimize a mesh to reduce discretization error. This paper compares the application of multiple solution adaptive techniques in combination with multiple flow solvers to solve for the flow field about a 2D airfoil section of the NASA High-Lift Common Research Model (HL-CRM). By driving the adaptive mesh processes to a similar level of mesh convergence, the ability to achieve consistent results between multiple adaptive techniques and flow solvers is demonstrated. Mesh convergence for the various adaptive mesh approaches is compared identifying potential areas for improvement and providing mesh generation guidance for future workshops.

mesh adaptation↗
NASA NTRS1988

Error indicators and accuracy improvements of finite element solutions

Practical and reliable estimators of the discretization errors in engineering problems are developed. Error indicators for identifying the regions or elements of the solution domain which are likely to have the largest discretization errors are presented, and a simple computational procedure for improving the accuracy of the finite element solutions for shell problems is given. The similarities between the proposed procedure and a preconditioned conjugate gradient (PCG) technique are identified and exploited to generate pointwise error indicators from the PCG technique. Numerical examples in the linear static analysis of shells are presented.

Noor, Ahmed K.↗
NASA NTRS2004

Sampling Versus Filtering in Large-Eddy Simulations

A LES formalism in which the filter operator is replaced by a sampling operator is proposed. The unknown quantities that appear in the LES equations originate only from inadequate resolution (Discretization errors). The resulting viewpoint seems to make a link between finite difference approaches and finite element methods. Sampling operators are shown to commute with nonlinearities and to be purely projective. Moreover, their use allows an unambiguous definition of the LES numerical grid. The price to pay is that sampling never commutes with spatial derivatives and the commutation errors must be modeled. It is shown that models for the discretization errors may be treated using the dynamic procedure. Preliminary results, using the Smagorinsky model, are very encouraging.

Debliquy, O.↗
NASA NTRS2018

Unstructured Grid Adaptation and Solver Technology for Turbulent Flows

Unstructured grid adaptation is a tool to control Computational Fluid Dynamics (CFD) discretization error. However, adaptive grid techniques have made limited impact on production analysis workflows where the control of discretization error is critical to obtaining reliable simulation results. Issues that prevent the use of adaptive grid methods are identified by applying unstructured grid adaptation methods to a series of benchmark cases. Once identified, these challenges to existing adaptive workflows can be addressed. Unstructured grid adaptation is evaluated for test cases described on the Turbulence Modeling Resource (TMR) web site, which documents uniform grid refinement of multiple schemes. The cases are turbulent flow over a Hemisphere Cylinder and an ONERA M6Wing. Adaptive grid force and moment trajectories are shown for three integrated grid adaptation processes with Mach interpolation control and output error based metrics. The integrated grid adaptation process with a finite element (FE) discretization produced results consistent with uniform grid refinement of fixed grids. The integrated grid adaptation processes with finite volume schemes were slower to converge to the reference solution than the FE method. Metric conformity is documented on grid/metric snapshots for five grid adaptation mechanics implementations. These tools produce anisotropic boundary conforming grids requested by the adaptation process.

Park, Michael A.↗
NASA NTRS1986

Self-adaptive difference method for the effective solution of computationally complex problems of boundary layer theory

An implicit difference procedure for the solution of equations for a chemically reacting hypersonic boundary layer is described. Difference forms of arbitrary error order in the x and y coordinate plane were used to derive estimates for discretization error. Computational complexity and time were minimized by the use of this difference method and the iteration of the nonlinear boundary layer equations was regulated by discretization error. Velocity and temperature profiles are presented for Mach 20.14 and Mach 18.5; variables are velocity profiles, temperature profiles, mass flow factor, Stanton number, and friction drag coefficient; three figures include numeric data.

Schoenauer, W.↗
NASA NTRS2010

Faster and More Accurate Transport Procedures for HZETRN

Several aspects of code verification are examined for HZETRN. First, a detailed derivation of the numerical marching algorithms is given. Next, a new numerical method for light particle transport is presented, and improvements to the heavy ion transport algorithm are discussed. A summary of various coding errors is also given, and the impact of these errors on exposure quantities is shown. Finally, a coupled convergence study is conducted. From this study, it is shown that past efforts in quantifying the numerical error in HZETRN were hindered by single precision calculations and computational resources. It is also determined that almost all of the discretization error in HZETRN is caused by charged target fragments below 50 AMeV. Total discretization errors are given for the old and new algorithms, and the improved accuracy of the new numerical methods is demonstrated. Run time comparisons are given for three applications in which HZETRN is commonly used. The new algorithms are found to be almost 100 times faster for solar particle event simulations and almost 10 times faster for galactic cosmic ray simulations.

Slaba, Tony C.↗
NASA NTRS2014

Toward Automatic Verification of Goal-Oriented Flow Simulations

We demonstrate the power of adaptive mesh refinement with adjoint-based error estimates in verification of simulations governed by the steady Euler equations. The flow equations are discretized using a finite volume scheme on a Cartesian mesh with cut cells at the wall boundaries. The discretization error in selected simulation outputs is estimated using the method of adjoint-weighted residuals. Practical aspects of the implementation are emphasized, particularly in the formulation of the refinement criterion and the mesh adaptation strategy. Following a thorough code verification example, we demonstrate simulation verification of two- and three-dimensional problems. These involve an airfoil performance database, a pressure signature of a body in supersonic flow and a launch abort with strong jet interactions. The results show reliable estimates and automatic control of discretization error in all simulations at an affordable computational cost. Moreover, the approach remains effective even when theoretical assumptions, e.g., steady-state and solution smoothness, are relaxed.

Simulations↗