Search NASA⌕ Search

SEARCH · Search NASA

Results for “generalized finite differences”

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 307 records · Page 17
All sourcesNASA NTRSDOE OSTI
NASA NTRS1978

CAVE3: A general transient heat transfer computer code utilizing eigenvectors and eigenvalues

The method of solution is a hybrid analytical numerical technique which utilizes eigenvalues and eigenvectors. The method is inherently stable, permitting large time steps even with the best of conductors with the finest of mesh sizes which can provide a factor of five reduction in machine time compared to conventional explicit finite difference methods when structures with small time constants are analyzed over long time periods. This code will find utility in analyzing hypersonic missile and aircraft structures which fall naturally into this class. The code is a completely general one in that problems involving any geometry, boundary conditions and materials can be analyzed. This is made possible by requiring the user to establish the thermal network conductances between nodes. Dynamic storage allocation is used to minimize core storage requirements. This report is primarily a user's manual for CAVE3 code. Input and output formats are presented and explained. Sample problems are included which illustrate the usage of the code as well as establish the validity and accuracy of the method.

Palmieri, J. V.↗
NASA NTRS1992

Two-dimensional hybrid continuum/particle approach for rarefied flows

A hybrid numerical technique previously developed for one-dimensional rarefied gas flows is generalized to two dimensions. The method is based on the fact that the flowfield that develops near a body in a rarefied gas typically contains local regions of continuum, transitional and free-molecular flow. By utilizing the solution technique most appropriate for each region and coupling the techniques in an interface region where both are applicable, more computationally efficient solutions can be obtained, or equivalently, more complex flowfields can be analyzed. The present method combines finite difference solution of the Navier-Stokes equations in the continuum regions, with Direct Simulation Monte Carlo in the more rarefied regions. The two schemes are coupled interactively via a general conservative flux boundary condition. The method is tested by application to the model problem of pressure-driven rarefied flow through a slit. Results show the hybrid scheme offers a speedup of a factor of nearly two for the nominal conditions considered, due to the decrease in the size of the Direct Simulation domain.

Wadsworth, Dean C.↗
NASA NTRS1976

Some recent progress in transonic flow computation

Although the development of a finite difference relaxation procedure to solve the steady form of equations of motion gave birth to the study of computational transonic aerodynamics and considerable progress has been made using the small disturbance theory, no general analytical solution method yet exists for transonic flows that include three dimensional unsteady, and viscous effects. Two techniques are described which are useful in computational transonic aerodynamics applications. The finite volume method simplifies the application of boundary conditions without introducing the constriction associated with small disturbance theory. Governing equations are solved in a Cartesian coordinate system using a body-oriented and shock-oriented mesh network. Only the volume and surface normal directions of the volume elements must be known. The other method, configuration design by numerical optimization, can be used by aircraft designers to develop configurations that satisfy specific geometric performance constraints. Two examples of airfoil design by numerical optimization are presented.

Ballhaus, W. F.↗
DOE OSTI2016

MPACT Theory Manual, Version 2.2.0

This theory manual describes the three-dimensional (3-D) whole-core, pin-resolved transport calculation methodology employed in the MPACT code. To provide sub-pin level power distributions with sufficient accuracy, MPACT employs the method of characteristics (MOC) solutions in the framework of a 3-D coarse mesh finite difference (CMFD) formulation. MPACT provides a 3D MOC solution, but also a 2D/1D solution in which the 2D planar solution is provided by MOC and the axial coupling is resolved by one-dimensional (1-D) lower order (diffusion or P3) solutions. In Chapter 2 of the manual, the MOC methodology is described for calculating the regional angular and scalar fluxes from the Boltzmann transport equation. In Chapter 3, the 2D/1D methodology is described, together with the description of the CMFD iteration process involving dynamic homogenization and solution of the multigroup CMFD linear system. A description of the MPACT depletion algorithm is given in Chapter 4, followed by a discussion of the subgroup and ESSM resonance processing methods in Chapter 5. The final Chapter 6 describes a simplified thermal hydraulics model in MPACT.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2019

Asymptotics of the General Solution of a Linear Singularly Perturbed System of Higher-Order Differential Equations with Degenerations

By using the theory of polynomial matrix pencils, we construct the asymptotics of linearly independent solutions of homogeneous singularly perturbed system of linear differential equations of any order m with matrices at higher derivatives degenerating as a small parameter approaches zero. The general case is analyzed. In this case, the limit matrix pencil has several finite and infinite elementary divisors of both the same and different multiplicities. The corresponding asymptotic estimates are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Sylvester-preconditioned adaptive-rank implicit time integrators for advection-diffusion equations with variable coefficients

Here, we consider the adaptive-rank integration of multi-dimensional time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with high-order diagonally implicit Runge-Kutta temporal schemes. The discrete equation is a generalized Sylvester equation (GSE), which we solve with a projection-based adaptive-rank algorithm structured around two key strategies: (i) constructing dimension-wise subspaces using a novel atypical extended Krylov strategy, and (ii) efficiently solving the basis coefficient matrix with a preconditioned GMRES solver. The low-rank decomposition is performed in 2D using SVD and with high-order SVD (HOSVD) in 3D to represent the tensor in a compressed Tucker format. For d-dimensional problems (here, d = 2 or 3), the computational complexity and memory storage of the approach are found numerically to scale as and $\mathscr{O}(Nr^2) + \mathscr{O} (r^{d+1})$ and $\mathscr{O}(Nr) + \mathscr{O} (r^{d})$, respectively, with the one-dimensional resolution and the maximal rank during the Krylov iteration (which we find to be largely independent of on our numerical examples). We present numerical examples that illustrate the advertised properties of the algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Efficient real space formalism for hybrid density functionals

We present an efficient real space formalism for hybrid exchange-correlation functionals in generalized Kohn–Sham density functional theory (DFT). In particular, we develop an efficient representation for any function of the real space finite-difference Laplacian matrix by leveraging its Kronecker product structure, thereby enabling the time to solution of associated linear systems to be highly competitive with the fast Fourier transform scheme while not imposing any restrictions on the boundary conditions. We implement this formalism for both the unscreened and range-separated variants of hybrid functionals. We verify its accuracy and efficiency through comparisons with established planewave codes for isolated as well as bulk systems. In particular, we demonstrate up to an order-of-magnitude speedup in time to solution for the real space method. We also apply the framework to study the structure of liquid water using ab initio molecular dynamics, where we find good agreement with the literature. Overall, the current formalism provides an avenue for efficient real-space DFT calculations with hybrid density functionals.

Chemistry↗
DOE OSTI2016

Implementation of a Red-Black SOR CMFD Solver in MPACT

As part of the Consortium for Advanced Simulation of Light Water Reactors (CASL), the Virtual Environment for Reactor Applications (VERA) is being developed to provide high-fidelity multi-physics simulations of nuclear reactor cores. The MPACT code being developed collaboratively by Oak Ridge National Laboratory and the University of Michigan includes the primary deterministic neutron transport solver available within VERA. MPACT employs the 2D/1D method to solve three-dimensional problems, using 2D method of characteristics (MOC) radial transport solvers with 1D NEM-SP3 axial transport solvers, which are coupled using axial and radial transverse leakages. A 3D coarse mesh finite difference (CMFD) solver is used to provide accelerated convergence that is more stable. The Portable, Extensible Tool kit for Scientific Computation (PETSc) generalized minimal residual (GMRES) iterative solver with a block Jacobi preconditioner has been used to solve the CMFD linear systems. This work focused on the addition of a red-black successive over-relaxation (RBSOR) solver to reduce overall runtime by reducing time spent solving the CMFD linear systems. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
NASA NTRS1987

Comparison of generalized Reynolds and Navier Stokes equations for flow of a power law fluid

This paper compares a finite element solution of a modified Reynolds equation with a finite difference solution of the Navier-Stokes equation for a power law fluid. Both the finite element and finite difference formulation are reviewed. Solutions to spiral flow in parallel and conical geometries are compared. Comparison with experimental results are also given. The effects of the assumptions used in the Reynolds equation are discussed.

Mullen, R. L.↗
DOE OSTI2017

Abnormal blocking of a guided mode propagating in a silicon optical waveguide with periodic tunnel coupling

This paper considers abnormal blocking of a guided mode propagating in a silicon optical waveguide with periodic tunnelling inserts. Using an independent two-dimensional analysis by the method of lines (MoL) and direct simulation by the finite-difference time-domain (FDTD) method, we have identified additional signal blocking bands, unrelated to Bragg conversion to backward guided modes of the parent silicon waveguide. These bands are due to the conversion of the incident wave energy to a leaky quasi-mode of the periodically segmented structure, which subsequently transfers the energy to the ambient medium in the form of radiation modes. A distinctive feature of this phenomenon is resonant coupling of the guided mode of the strip waveguide with its radiation modes, which is due to the weak tunnel coupling with the periodically segmented structure. This structure does not support independent guided propagation, so the energy stored in it is re-emitted to space. The abnormal blocking effect may find application in optical telecommunications elements and in the fabrication of optical sensors. (paper)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1993

Issac, Jason Cherian ses in transonic flow

Flutter analysis of a two degree of freedom airfoil in compressible flow is performed using a state-space representation of the unsteady aerodynamic behavior. Indicial response functions are used to represent the normal force and moment response of the airfoil. The structural equations of motion of the airfoil with bending and torsional degrees of freedom are coupled to the unsteady air loads and the aeroelastic system so modelled is solved as an eigenvalue problem to determine the stability. The aeroelastic equations are also directly integrated with respect to time and the time-domain results compared with the results from the eigenanalysis. A good agreement is obtained. The derivatives of the flutter speed obtained from the eigenanalysis are calculated with respect to the mass and stiffness parameters by both analytical and finite-difference methods for various transonic Mach numbers. The experience gained from the two degree of freedom model is applied to study the sensitivity of the flutter response of a wing with respect to various shape parameters. The parameters being considered are as follows: (1) aspect ratio; (2) surface area of the wing; (3) taper ratio; and (4) sweep. The wing deflections are represented by Chebyshev polynomials. The compressible aerodynamic state-space model used for the airfoil section is extended to represent the unsteady aerodynamic forces on a generally laminated tapered skewed wing. The aeroelastic equations are solved as an eigenvalue problem to determine the flutter speed of the wing. The derivatives of the flutter speed with respect to the shape parameters are calculated by both analytical and finite difference methods.

Issac, Jason Cherion↗
DOE OSTI2022

Simulating magnetized neutron stars with discontinuous Galerkin methods

Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard minmod /ΛΠ N limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized Tolman-Oppenheimer-Volkoff stars without any fine-tuning of parameters. Among the limiters explored in the paper, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1981

On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. II - Five-point schemes

This paper presents a family of two-level five-point implicit schemes for the solution of one-dimensional systems of hyperbolic conservation laws, which generalized the Crank-Nicholson scheme to fourth order accuracy (4-4) in both time and space. These 4-4 schemes are nondissipative and unconditionally stable. Special attention is given to the system of linear equations associated with these 4-4 implicit schemes. The regularity of this system is analyzed and efficiency of solution-algorithms is examined. A two-datum representation of these 4-4 implicit schemes brings about a compactification of the stencil to three mesh points at each time-level. This compact two-datum representation is particularly useful in deriving boundary treatments. Numerical results are presented to illustrate some properties of the proposed scheme.

Harten, A.↗
DOE OSTI2024

An Empirical Quantile Estimation Approach for Chance-Constrained Nonlinear Optimization Problems

We investigate an empirical quantile estimation approach to solve chance-constrained nonlinear optimization problems. Our approach is based on the reformulation of the chance constraint as an equivalent quantile constraint to provide stronger signals on the gradient. In this approach, the value of the quantile function is estimated empirically from samples drawn from the random parameters, and the gradient of the quantile function is estimated via a finite-difference approximation on top of the quantile-function-value estimation. We establish a convergence theory of this approach within the framework of an augmented Lagrangian method for solving general nonlinear constrained optimization problems. The foundation of the convergence analysis is a concentration property of the empirical quantile process, and the analysis is divided based on whether or not the quantile function is differentiable. In contrast to the sampling-and-smoothing approach used in the literature, the method developed in this paper does not involve any smoothing function and hence the quantile-function gradient approximation is easier to implement and there are less accuracy-control parameters to tune. Furthermore, we demonstrate the effectiveness of this approach and compare it with a smoothing method for the quantile-gradient estimation. Numerical investigation shows that the two approaches are competitive for certain problem instances.

Applied Probability↗
NASA NTRS1984

Parameter studies of gear cooling using an automatic finites element mesh generator

The range of accuracies achieved in the gear tooth temperature using an automatic finite element mesh generator were investigated. Gear web contribution to the gear cooling process was studied by introducing a varying size hole at the center of the gear because of the versatility of program TARG in allowing different heat transfer coefficients in different areas of the gear tooth. A study was carried out to evaluate the contribution of the loaded and unloaded faces as well as the top and bottom lands. A general purpose two-dimensional finite element preprocessor ATOGEN has been developed for automatic generation of a finite element mesh over a pie-shaped sector of a gear. The program was used for facilitating the input to an upgraded version of a previously developed program for the thermal analysis of running gears (TARG). The latter program determined the steady state temperature distribution throughout the specified gear. The automatic mesh generator program includes a band width minimization routine for reducing computer cost.

El-Bayoumy, L. E.↗
DOE OSTI2026

Regional surrogates for predictive control of digital twins

Digital twins of complex systems must involve a model that is fast, generalizable, and usable for real-time control. For example, high-fidelity nonlinear multiphysics simulations can capture laser-material interactions, but are too slow for optimization or model predictive control (MPC). Reduced-order models, used to accelerate such computation, frequently fail to generalize to unseen inputs or control states. We show theoretically that this failure is intrinsic, i.e., that a learned model is non-unique outside the sampled subspace when its low-rank structure arises from limited excitation and clustered eigenvalues, rather than from a user-imposed truncation alone. Motivated by this result, we propose a control-ready regional surrogate-construction framework for both autonomous and nonautonomous dynamics; it employs Koopman lifting to represent nonlinearities, while preserving spatial locality. We illustrate our approach by constructing a control-ready surrogate for the digital twin of a thermal component of additive-manufacturing process. Our surrogate, localized in space through a von Neumann stencil, is learned from noisy high-fidelity simulations that emulate thermal-camera images collected during the manufacturing. It is linear in thermo-physically augmented states so that MPC reduces to a convex quadratic program. The surrogate requires no online correction, generalizes to unseen scan paths and power profiles of the laser, and is more than three orders of magnitude faster than a finite-difference solver. Furthermore, when the MPC sequence computed on the digital twin is applied to this solver, closed-loop temperature regulation is recovered, showing that the surrogate preserves control-relevant input-output behavior.

Data-driven model↗
DOE OSTI2017

Peakompactons: Peaked compact nonlinear waves

This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed peakompactons. We present that these peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg–de Vries-type models. Peakompactons, like the now-well-known compactons and unlike the soliton solutions of the Korteweg–de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, i.e., a higher spatial derivative suffers a jump discontinuity at the wave’s crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. Lastly, a simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solutions, the general physical features of the so-called K # (n,m) hierarchy of nonlinearly dispersive Korteweg–de Vries-type models are discussed as well.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1982

Influence of land-surface evapotranspiration on the earth's climate

Land-surface evapotranspiration is shown to strongly influence global fields of rainfall, temperature and motion by calculations using a numerical model of the atmosphere, confirming the general belief in the importance of evapotranspiration-producing surface vegetation for the earth's climate. The current version of the Goddard Laboratory atmospheric general circulation model is used in the present experiment, in which conservation equations for mass, momentum, moisture and energy are expressed in finite-difference form for a spherical grid to calculate (1) surface pressure field evolution, and (2) the wind, temperature, and water vapor fields at nine levels between the surface and a 20 km height.

Shukla, J.↗