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NASA NTRS1994

Accurate finite difference methods for time-harmonic wave propagation

Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.

Harari, Isaac↗

NASA NTRS1994

Preliminary estimates of nucleon fluxes in a water target exposed to solar-flare protons: BRYNTRN versus Monte Carlo code

A baryon transport code (BRYNTRN) has previously been verified using available Monte Carlo results for a solar-flare spectrum as the reference. Excellent results were obtained, but the comparisons were limited to the available data on dose and dose equivalent for moderate penetration studies that involve minor contributions from secondary neutrons. To further verify the code, the secondary energy spectra of protons and neutrons are calculated using BRYNTRN and LAHET (Los Alamos High-Energy Transport code, which is a Monte Carlo code). These calculations are compared for three locations within a water slab exposed to the February 1956 solar-proton spectrum. Reasonable agreement was obtained when various considerations related to the calculational techniques and their limitations were taken into account. Although the Monte Carlo results are preliminary, it appears that the neutron albedo, which is not currently treated in BRYNTRN, might be a cause for the large discrepancy seen at small penetration depths. It also appears that the nonelastic neutron production cross sections in BRYNTRN may underestimate the number of neutrons produced in proton collisions with energies below 200 MeV. The notion that the poor energy resolution in BRYNTRN may cause a large truncation error in neutron elastic scattering requires further study.

Shinn, Judy L.↗

NASA NTRS1995

A compact solution to computational acoustics

This paper demonstrates that the linearized, dimensional Euler equations for acoustic computation can be accurately solved as a set of decoupled first-order wave equations, and that if ordered properly, this system of simple waves has unambiguous, easily implemented boundary conditions, allowing waves of same group speeds to pass through numerical boundaries or comply with wall conditions. Thus, the task of designing a complex multi-dimensional scheme with approximate far-field boundary conditions reduces to the design of higher order schemes for the one-dimensional simple wave equation. A compact finite-difference scheme and a characteristically exact but numerically n(th) order accurate boundary condition are introduced for solving the first order wave equation. Spanning a three-point two-level stencil, this low-dispersion implicit scheme has a third order spatial accuracy when used on nonuniform meshes, fourth order accurate on uniform meshes, and a temporal accuracy of second order due to the choice of trapezoidal integration for algorithmic simplicity. The robustness and accuracy of the scheme are demonstrated through a series of numerical experiments and comparisons with published results. When tested on the one-dimensional wave equation on a uniform grid, this scheme allows a Gaussian wave packet to pass through any finite domain with low numerical dispersion characteristic of a spatially fourth-order scheme and reflections at numerical boundaries maintained below truncation error. On highly stretched and irregular grids, only mild dispersions are found in the solution while solutions by other methods fail or are severely distorted. Yet, this scheme is no more sophisticated to solve or implement than the Crank-Nicolson scheme. This scheme has been tested on four categories of the ICASE/LaRC benchmark problems, which include propagation of acoustic and convective waves in Cartesian and cylindrical domains, reflection of acoustic wave at stationary/moving boundaries, and sound generation by gust-blade interaction.

Fung, K.-Y.↗

NASA NTRS1995

Nonstandard finite difference schemes

The major research activities of this proposal center on the construction and analysis of nonstandard finite-difference schemes for ordinary and partial differential equations. In particular, we investigate schemes that either have zero truncation errors (exact schemes) or possess other significant features of importance for numerical integration. Our eventual goal is to bring these methods to bear on problems that arise in the modeling of various physical, engineering, and technological systems. At present, these efforts are extended in the direction of understanding the exact nature of these nonstandard procedures and extending their use to more complicated model equations. Our presentation will give a listing (obtained to date) of the nonstandard rules, their application to a number of linear and nonlinear, ordinary and partial differential equations. In certain cases, numerical results will be presented.

Mickens, Ronald E.↗

NASA NTRS1995

An accuracy assessment of Cartesian-mesh approaches for the Euler equations

A critical assessment of the accuracy of Cartesian-mesh approaches for steady, transonic solutions of the Euler equations of gas dynamics is made. An exact solution of the Euler equations (Ringleb's flow) is used not only to infer the order of the truncation error of the Cartesian-mesh approaches, but also to compare the magnitude of the discrete error directly to that obtained with a structured mesh approach. Uniformly and adaptively refined solutions using a Cartesian-mesh approach are obtained and compared to each other and to uniformly refined structured mesh results. The effect of cell merging is investigated as well as the use of two different K-exact reconstruction procedures. The solution methodology of the schemes is explained and tabulated results are presented to compare the solution accuracies.

Coirier, William J.↗

NASA NTRS1996

Adaptive grid methods for RLV environment assessment and nozzle analysis

Rapid access to highly accurate data about complex configurations is needed for multi-disciplinary optimization and design. In order to efficiently meet these requirements a closer coupling between the analysis algorithms and the discretization process is needed. In some cases, such as free surface, temporally varying geometries, and fluid structure interaction, the need is unavoidable. In other cases the need is to rapidly generate and modify high quality grids. Techniques such as unstructured and/or solution-adaptive methods can be used to speed the grid generation process and to automatically cluster mesh points in regions of interest. Global features of the flow can be significantly affected by isolated regions of inadequately resolved flow. These regions may not exhibit high gradients and can be difficult to detect. Thus excessive resolution in certain regions does not necessarily increase the accuracy of the overall solution. Several approaches have been employed for both structured and unstructured grid adaption. The most widely used involve grid point redistribution, local grid point enrichment/derefinement or local modification of the actual flow solver. However, the success of any one of these methods ultimately depends on the feature detection algorithm used to determine solution domain regions which require a fine mesh for their accurate representation. Typically, weight functions are constructed to mimic the local truncation error and may require substantial user input. Most problems of engineering interest involve multi-block grids and widely disparate length scales. Hence, it is desirable that the adaptive grid feature detection algorithm be developed to recognize flow structures of different type as well as differing intensity, and adequately address scaling and normalization across blocks. These weight functions can then be used to construct blending functions for algebraic redistribution, interpolation functions for unstructured grid generation, forcing functions to attract/repel points in an elliptic system, or to trigger local refinement, based upon application of an equidistribution principle. The popularity of solution-adaptive techniques is growing in tandem with unstructured methods. The difficultly of precisely controlling mesh densities and orientations with current unstructured grid generation systems has driven the use of solution-adaptive meshing. Use of derivatives of density or pressure are widely used for construction of such weight functions, and have been proven very successful for inviscid flows with shocks. However, less success has been realized for flowfields with viscous layers, vortices or shocks of disparate strength. It is difficult to maintain the appropriate mesh point spacing in the various regions which require a fine spacing for adequate resolution. Mesh points often migrate from important regions due to refinement of dominant features. An example of this is the well know tendency of adaptive methods to increase the resolution of shocks in the flowfield around airfoils, but in the incorrect location due to inadequate resolution of the stagnation region. This problem has been the motivation for this research.

Thornburg, Hugh J.↗

NASA NTRS1996

An Algebraic Multigrid Solver for Navier-Stokes Problems in the Discrete Second-Order Approximation

An algebraic multigrid scheme is presented for solving the discrete Navier-Stokes equations to second-order accuracy using the defect-correction method. Solutions have been obtained for problems involving both structured and unstructured meshes, with the resolution and resolution grading controlled by global and local mesh refinements. The solver is efficient and robust to the extent that no underrelaxation of variables has been required to ensure convergence, but rates of convergence can be improved with small amounts of underrelaxation of the velocity-pressure coupling. Provided that the computational mesh can resolve the flow field, convergence characteristics are almost mesh independent. Rates of convergence actually improve with refinement, asymptotically approaching mesh independent values. For extremely coarse meshes where dispersive truncation errors would be expected to prevent convergence (or even induce divergence), solutions can still be obtained by using explicit underrelaxation in the iterative cycle.

Webster, R.↗

NASA NTRS1996

A Full Multi-Grid Method for the Solution of the Cell Vertex Finite Volume Cauchy-Riemann Equations

The system of inhomogeneous Cauchy-Riemann equations defined on a square domain and subject to Dirichlet boundary conditions is considered. This problem is discretised by using the cell vertex finite volume method on quadrilateral meshes. The resulting algebraic problem is overdetermined and the solution is defined in a least squares sense. By this approach a consistent algebraic problem is obtained which differs from the original one by O(h(exp 2)) perturbations of the right-hand side. A suitable cell-based convergent smoothing iteration is presented which is naturally linked to the least squares formulation. Hence, a standard multi-grid algorithm is reported which combines the given smoother and cell-based transfer operators. Some remarkable reduction properties of these operators are shown. A full multi-grid method is discussed which solves the discrete problem to the level of truncation error by employing one multi-grid cycle at each current level of discretisation. Experiments and applications of the full multi-grid scheme are presented.

Borzi, A.↗

NASA NTRS1998

Helioseismic Constraints on New Solar Models from the MoSEC Code

Evolutionary solar models are computed using a new stellar evolution code, MOSEC (Modular Stellar Evolution Code). This code has been designed with carefully controlled truncation errors in order to achieve a precision which reflects the increasingly accurate determination of solar interior structure by helioseismology. A series of models is constructed to investigate the effects of the choice of equation of state (OPAL or MHD-E, the latter being a version of the MHD equation of state recalculated by the author), the inclusion of helium and heavy-element settling and diffusion, and the inclusion of a simple model of mixing associated with the solar tachocline. The neutrino flux predictions are discussed, while the sound speed of the computed models is compared to that of the sun via the latest inversion of SOI-NMI p-mode frequency data. The comparison between models calculated with the OPAL and MHD-E equations of state is particularly interesting because the MHD-E equation of state includes relativistic effects for the electrons, whereas neither MHD nor OPAL do. This has a significant effect on the sound speed of the computed model, worsening the agreement with the solar sound speed. Using the OPAL equation of state and including the settling and diffusion of helium and heavy elements produces agreement in sound speed with the helioseismic results to within about +.-0.2%; the inclusion of mixing slightly improves the agreement.

Elliott, J. R.↗

NASA NTRS2001

Automatic Overset Grid Generation with Heuristic Feedback Control

An advancing front grid generation system for structured Overset grids is presented which automatically modifies Overset structured surface grids and control lines until user-specified grid qualities are achieved. The system is demonstrated on two examples: the first refines a space shuttle fuselage control line until global truncation error is achieved; the second advances, from control lines, the space shuttle orbiter fuselage top and fuselage side surface grids until proper overlap is achieved. Surface grids are generated in minutes for complex geometries. The system is implemented as a heuristic feedback control (HFC) expert system which iteratively modifies the input specifications for Overset control line and surface grids. It is developed as an extension of modern control theory, production rules systems and subsumption architectures. The methodology provides benefits over the full knowledge lifecycle of an expert system for knowledge acquisition, knowledge representation, and knowledge execution. The vector/matrix framework of modern control theory systematically acquires and represents expert system knowledge. Missing matrix elements imply missing expert knowledge. The execution of the expert system knowledge is performed through symbolic execution of the matrix algebra equations of modern control theory. The dot product operation of matrix algebra is generalized for heuristic symbolic terms. Constant time execution is guaranteed.

Robinson, Peter I.↗

NASA NTRS2002

Progress in the Simulation of Steady and Time-Dependent Flows with 3D Parallel Unstructured Cartesian Methods

The proposed paper will present recent extensions in the development of an efficient Euler solver for adaptively-refined Cartesian meshes with embedded boundaries. The paper will focus on extensions of the basic method to include solution adaptation, time-dependent flow simulation, and arbitrary rigid domain motion. The parallel multilevel method makes use of on-the-fly parallel domain decomposition to achieve extremely good scalability on large numbers of processors, and is coupled with an automatic coarse mesh generation algorithm for efficient processing by a multigrid smoother. Numerical results are presented demonstrating parallel speed-ups of up to 435 on 512 processors. Solution-based adaptation may be keyed off truncation error estimates using tau-extrapolation or a variety of feature detection based refinement parameters. The multigrid method is extended to for time-dependent flows through the use of a dual-time approach. The extension to rigid domain motion uses an Arbitrary Lagrangian-Eulerlarian (ALE) formulation, and results will be presented for a variety of two- and three-dimensional example problems with both simple and complex geometry.

Aftosmis, M. J.↗

NASA NTRS1999

An O(Nm(sup 2)) Plane Solver for the Compressible Navier-Stokes Equations

A hierarchical multigrid algorithm for efficient steady solutions to the two-dimensional compressible Navier-Stokes equations is developed and demonstrated. The algorithm applies multigrid in two ways: a Full Approximation Scheme (FAS) for a nonlinear residual equation and a Correction Scheme (CS) for a linearized defect correction implicit equation. Multigrid analyses which include the effect of boundary conditions in one direction are used to estimate the convergence rate of the algorithm for a model convection equation. Three alternating-line- implicit algorithms are compared in terms of efficiency. The analyses indicate that full multigrid efficiency is not attained in the general case; the number of cycles to attain convergence is dependent on the mesh density for high-frequency cross-stream variations. However, the dependence is reasonably small and fast convergence is eventually attained for any given frequency with either the FAS or the CS scheme alone. The paper summarizes numerical computations for which convergence has been attained to within truncation error in a few multigrid cycles for both inviscid and viscous ow simulations on highly stretched meshes.

Thomas, J. L.↗

NASA NTRS1994

Space proton transport in one dimension

An approximate evaluation procedure is derived for a second-order theory of coupled nucleon transport in one dimension. An analytical solution with a simplified interaction model is used to determine quadrature parameters to minimize truncation error. Effects of the improved method on transport solutions with the BRYNTRN data base are evaluated. Comparisons with Monte Carlo benchmarks are given. Using different shield materials, the computational procedure is used to study the physics of space protons. A transition effect occurs in tissue near the shield interface and is most important in shields of high atomic number.

NASA Discipline Number 45-10↗

NASA NTRS2008

High-Order Energy Stable WENO Schemes

A new third-order Energy Stable Weighted Essentially NonOscillatory (ESWENO) finite difference scheme for scalar and vector linear hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. A rigorous truncation error analysis is presented showing that the accuracy of the 3rd-order ESWENO scheme is drastically improved if the tuning parameters of the weight functions satisfy certain criteria. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO finite difference scheme of Jiang and Shu in terms of accuracy, while providing essentially nonoscillatory solutions near strong discontinuities.

Yamaleev, Nail K.↗

NASA NTRS2000

A Domain-Decomposed Multilevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries

Preliminary verification and validation of an efficient Euler solver for adaptively refined Cartesian meshes with embedded boundaries is presented. The parallel, multilevel method makes use of a new on-the-fly parallel domain decomposition strategy based upon the use of space-filling curves, and automatically generates a sequence of coarse meshes for processing by the multigrid smoother. The coarse mesh generation algorithm produces grids which completely cover the computational domain at every level in the mesh hierarchy. A series of examples on realistically complex three-dimensional configurations demonstrate that this new coarsening algorithm reliably achieves mesh coarsening ratios in excess of 7 on adaptively refined meshes. Numerical investigations of the scheme's local truncation error demonstrate an achieved order of accuracy between 1.82 and 1.88. Convergence results for the multigrid scheme are presented for both subsonic and transonic test cases and demonstrate W-cycle multigrid convergence rates between 0.84 and 0.94. Preliminary parallel scalability tests on both simple wing and complex complete aircraft geometries shows a computational speedup of 52 on 64 processors using the run-time mesh partitioner.

Aftosmis, M. J.↗

NASA NTRS2013

Spurious Behavior of Shock-Capturing Methods: Problems Containing Stiff Source Terms and Discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.

Yee, Helen M. C.↗

NASA NTRS2019

Quantifying Water Storage Change and Land Subsidence Induced by Reservoir Impoundment Using GRACE, Landsat, and GPS Data

The construction of hydropower dams is a common strategy to support a country's increasing need for electricity and river water management for industry and agriculture. Although the hydrological and geophysical impacts of water relocation are usually assessed prior to impoundment, their accuracy is generally limited due to the lack of in situ observations, especially in a remote area. This study presents a workflow to quantify the terrestrial water storage change (TWS) and land subsidence induced by a reservoir's water impoundment using multiple satellite observations (GRACE, Landsat), land surface models (CABLE, GLDAS, NCEP, ECMWF), and GPS data. The study site is the Bakun Dam, located in Sarawak, Malaysia, which is the largest hydropower dam in Southeast Asia. Commencing operation in late 2010, the dam induced a change of water mass and lake surface area that was clearly observed by GRACE and Landsat observations, respectively. During the 17-month impounding period (from August 2010 to December 2011), GRACE observed a dramatic increase of approximately 200 mm equivalent water height, while Landsat detected an increased lake extent of around 600 km2. In this paper, a forward model is developed to determine the increased water surface level corresponding to GRACE observations, estimated to be about 120 m. In contrast to GRACE, the TWS derived from land surface models cannot capture the increased TWS, due to the lack of reservoir routing algorithms in the models. In addition, the land subsidence was calculated using the disk load model constructed based on the GRACE-derived lake level and Landsat-derived lake extent; the result is validated with the GPS data from BIN1 station, located at the western coast of Borneo. The commencement stage of the Bakun Dam induces the large-scale land subsidence, which causes the GPS-BIN1 station to subside by ~9 mm, and move toward the Bakun Lake by ~4 mm. Computation of the surface displacements directly from GRACE spherical harmonic coefficient data fails to capture the subsidence feature, mainly due to the truncation error. Overall, this study demonstrates that evaluating GRACE in conjunction with Landsat, LSMs, and GPS data allows the exploitation of the gravity signal at a much smaller spatial scale than its intrinsic resolution. Benefiting from global coverage, the newly developed satellite-based algorithm is a valuable tool for assessing the impacts of reservoir operation on hydrological and geophysical changes from local to regional scales.

Natthachet Tangdamrongsub↗

NASA NTRS2019

Quantifying Water Storage Change and Land Subsidence Induced by Reservoir Impoundment Using GRACE, Landsat, and GPS Data

The construction of hydropower dams is a common strategy to support a country's increasing need for electricityand river water management for industry and agriculture. Although the hydrological and geophysical impacts ofwater relocation are usually assessed prior to impoundment, their accuracy is generally limited due to the lack ofin situ observations, especially in a remote area. This study presents a workflow to quantify the terrestrial waterstorage change (TWS) and land subsidence induced by a reservoir's water impoundment using multiple satelliteobservations (GRACE, Landsat), land surface models (CABLE, GLDAS, NCEP, ECMWF), and GPS data. The studysite is the Bakun Dam, located in Sarawak, Malaysia, which is the largest hydropower dam in Southeast Asia.Commencing operation in late 2010, the dam induced a change of water mass and lake surface area that wasclearly observed by GRACE and Landsat observations, respectively. During the 17-month impounding period(from August 2010 to December 2011), GRACE observed a dramatic increase of approximately 200mmequivalent water height, while Landsat detected an increased lake extent of around 600 km2. In this paper, aforward model is developed to determine the increased water surface level corresponding to GRACE observations,estimated to be about 120 m. In contrast to GRACE, the TWS derived from land surface models cannotcapture the increased TWS, due to the lack of reservoir routing algorithms in the models. In addition, the landsubsidence was calculated using the disk load model constructed based on the GRACE-derived lake level andLandsat-derived lake extent; the result is validated with the GPS data from BIN1 station, located at the westerncoast of Borneo. The commencement stage of the Bakun Dam induces the large-scale land subsidence, whichcauses the GPS-BIN1 station to subside by ~9 mm, and move toward the Bakun Lake by ~4 mm. Computation ofthe surface displacements directly from GRACE spherical harmonic coefficient data fails to capture the subsidencefeature, mainly due to the truncation error. Overall, this study demonstrates that evaluating GRACE inconjunction with Landsat, LSMs, and GPS data allows the exploitation of the gravity signal at a much smallerspatial scale than its intrinsic resolution. Benefiting from global coverage, the newly developed satellite-basedalgorithm is a valuable tool for assessing the impacts of reservoir operation on hydrological and geophysicalchanges from local to regional scales.

Tangdamrongsub, Natthachet↗