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NASA NTRS1994

Global Asymptotic Behavior of Iterative Implicit Schemes

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Yee, H. C.↗

NASA NTRS1996

First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H(exp 1) product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity with estimates that are uniform in the Lame constants.

Cai, Z.↗

NASA NTRS1994

The Dynamics of Some Iterative Implicit Schemes

Yee, H. C.↗

NASA NTRS2004

Putting Priors in Mixture Density Mercer Kernels

This paper presents a new methodology for automatic knowledge driven data mining based on the theory of Mercer Kernels, which are highly nonlinear symmetric positive definite mappings from the original image space to a very high, possibly infinite dimensional feature space. We describe a new method called Mixture Density Mercer Kernels to learn kernel function directly from data, rather than using predefined kernels. These data adaptive kernels can en- code prior knowledge in the kernel using a Bayesian formulation, thus allowing for physical information to be encoded in the model. We compare the results with existing algorithms on data from the Sloan Digital Sky Survey (SDSS). The code for these experiments has been generated with the AUTOBAYES tool, which automatically generates efficient and documented C/C++ code from abstract statistical model specifications. The core of the system is a schema library which contains template for learning and knowledge discovery algorithms like different versions of EM, or numeric optimization methods like conjugate gradient methods. The template instantiation is supported by symbolic- algebraic computations, which allows AUTOBAYES to find closed-form solutions and, where possible, to integrate them into the code. The results show that the Mixture Density Mercer-Kernel described here outperforms tree-based classification in distinguishing high-redshift galaxies from low- redshift galaxies by approximately 16% on test data, bagged trees by approximately 7%, and bagged trees built on a much larger sample of data by approximately 2%.

Srivastava, Ashok N.↗

NASA NTRS1991

Asymptotic expansions of the hydrogenic radial dipole integral

A method is presented for an asymptotic expansion of the hydrogenic radial dipole integral for discrete-discrete transitions from initial state n'l' to final state nl. The advantage of this simple method is that it allows all the necessary algebraic manipulations to be done by the computer programming package REDUCE-2. Expansion coefficients up to the coefficient of the l/n17 term are obtained for all initial states with n' < or = 10. However, only the coefficients which are not calculated by Klarsfeld are tabulated in this paper.

NASA Discipline Radiation Health↗

NASA NTRS2015

A Numerical and Experimental Study of Coflow Laminar Diffusion Flames: Effects of Gravity and Inlet Velocity

In this work, the influence of gravity, fuel dilution, and inlet velocity on the structure, stabilization, and sooting behavior of laminar coflow methane-air diffusion flames was investigated both computationally and experimentally. A series of flames measured in the Structure and Liftoff in Combustion Experiment (SLICE) was assessed numerically under microgravity and normal gravity conditions with the fuel stream CH4 mole fraction ranging from 0.4 to 1.0. Computationally, the MC-Smooth vorticity-velocity formulation of the governing equations was employed to describe the reactive gaseous mixture; the soot evolution process was considered as a classical aerosol dynamics problem and was represented by the sectional aerosol equations. Since each flame is axisymmetric, a two-dimensional computational domain was employed, where the grid on the axisymmetric domain was a nonuniform tensor product mesh. The governing equations and boundary conditions were discretized on the mesh by a nine-point finite difference stencil, with the convective terms approximated by a monotonic upwind scheme and all other derivatives approximated by centered differences. The resulting set of fully coupled, strongly nonlinear equations was solved simultaneously using a damped, modified Newton's method and a nested Bi-CGSTAB linear algebra solver. Experimentally, the flame shape, size, lift-off height, and soot temperature were determined by flame emission images recorded by a digital camera, and the soot volume fraction was quantified through an absolute light calibration using a thermocouple. For a broad spectrum of flames in microgravity and normal gravity, the computed and measured flame quantities (e.g., temperature profile, flame shape, lift-off height, and soot volume fraction) were first compared to assess the accuracy of the numerical model. After its validity was established, the influence of gravity, fuel dilution, and inlet velocity on the structure, stabilization, and sooting tendency of laminar coflow methane-air diffusion flames was explored further by examining quantities derived from the computational results.

microgravity↗

NASA NTRSDate not supplied

Matrix Theory for Data Association in PVS

Consider a collection of data generated by sensors from a set of aircraft. Data association is the process of connecting each sensor measurement with its corresponding aircraft. Furthermore once the data association has taken place, the state of the aircraft can be approximated using a Kalman filter. This talk aims to explore formal specification and verification of data association in the Prototype Verification System (PVS). Formal specification and verification of data association includes development of Kalman filters, Mahalanobis distance, and other topics of matrix analysis in PVS.

Linear Algebra↗

NASA NTRSDate not supplied

Instabilities in the Wake of Roughness on a Flat Plate in a Quiet Supersonic Tunnel

Roughness-induced transition is an unavoidable reality in practical high-speed vehicles. Typical prediction of transition due to roughness include algebraic correlations and, more recently, semi-empirical methods. In the NASA Langley Research Center Supersonic Low Disturbance Tunnel, a Mach 3.5 quiet tunnel, several transition experiments have been performed in the past decade to better understand the mechanisms by which small roughness causes transition in a supersonic boundary layer. The study started first with isolated roughness elements of different planforms and shapes and progressed to increasingly more complicated geometries before arriving at a pseudorandom roughness, defined by an analytic function. Concurrent computational efforts progressed with these studies as well, starting with the use of linear stability theory and progressing to the use of harmonic linearized Navier-Stokes to predict the growth of boundary layer stabilities.

Amanda Chou↗

DOE OSTI2022

Approximate symmetries in d = 4 CFTs with an Einstein gravity dual

By applying the stress-tensor-scalar operator product expansion (OPE) twice, we search for algebraic structures in d = 4 conformal field theories (CFTs) with a pure Einstein gravity dual. We find that a rescaled mode operator defined by an integral of the stress tensor T ++ on a d = 2 plane satisfies a Virasoro-like algebra when the dimension of the scalar is large. The structure is enhanced to include a Kac-Moody-type algebra if we incorporate the T -- component. In our scheme, the central terms are finite. It remains challenging to directly compute the stress-tensor sector of d = 4 scalar four-point functions at large central charge, which, based on holography and bootstrap methods, were recently shown to have a Virasoro/ $\mathcal{W}$-algebra vacuum block-like structure.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1998

Prediction of Very High Reynolds Number Compressible Skin Friction

Flat plate skin friction calculations over a range of Mach numbers from 0.4 to 3.5 at Reynolds numbers from 16 million to 492 million using a Navier Stokes method with advanced turbulence modeling are compared with incompressible skin friction coefficient correlations. The semi-empirical correlation theories of van Driest; Cope; Winkler and Cha; and Sommer and Short T' are used to transform the predicted skin friction coefficients of solutions using two algebraic Reynolds stress turbulence models in the Navier-Stokes method PAB3D. In general, the predicted skin friction coefficients scaled well with each reference temperature theory though, overall the theory by Sommer and Short appeared to best collapse the predicted coefficients. At the lower Reynolds number 3 to 30 million, both the Girimaji and Shih, Zhu and Lumley turbulence models predicted skin-friction coefficients within 2% of the semi-empirical correlation skin friction coefficients. At the higher Reynolds numbers of 100 to 500 million, the turbulence models by Shih, Zhu and Lumley and Girimaji predicted coefficients that were 6% less and 10% greater, respectively, than the semi-empirical coefficients.

Carlson, John R.↗

NASA NTRS1992

A new method for transonic static aeroelasticity problems

A new method has been developed to calculate the steady flow and structural deformations for fluid/structure interaction problems. The discretized fluid dynamic and structural equations are regarded as a single set of coupled, nonlinear, algebraic equations. The equilibrium solution is directly obtained using Newton's method. The governing equations used for the fluid flow are the two-dimensional Navier-Stokes equations, and a finite-element model is used to represent the structure. This paper describes the analytical method and presents sample calculations demonstrating the technique. The results show rapid convergence and good agreement with experimental data.

Felker, Fort F.↗

NASA NTRS1996

Chebyshev polynomials in the spectral Tau method and applications to Eigenvalue problems

Chebyshev Spectral methods have received much attention recently as a technique for the rapid solution of ordinary differential equations. This technique also works well for solving linear eigenvalue problems. Specific detail is given to the properties and algebra of chebyshev polynomials; the use of chebyshev polynomials in spectral methods; and the recurrence relationships that are developed. These formula and equations are then applied to several examples which are worked out in detail. The appendix contains an example FORTRAN program used in solving an eigenvalue problem.

Johnson, Duane↗

NASA NTRS1989

Aspects of efficient and reliable multibody system simulation

Multibody system equations can be generated in various forms. All of these may be interpreted as results of two basic approaches, the augmentation- and the elimination-method. The former method yields the descriptor form of the system motion, a set of differential-algebraic equations (DAE), and the latter the state space representation, a minimal set of ordinary differential equations (ODE). Both of these methods are surveyed. Particular emphasis is on the discussion of recursive computational schemes, generating the equations of motion with a number of operations, which is proportional to the number N of system bodies (O(N)-formulations). For simulation purposes one would like to create that set of system equations, which can be generated most efficiently and for which the most efficient and reliable solution techniques are available. Numerical solution techniques for ODE have been studied in great detail and they are well-developed. By contrast, DAE have not been investigated for such a long time. In view of new developments in the latter field the generation of all the equations required for an efficient and reliable solution of DAE describing multibody system motion is discussed. These methods, i.e., an O(N)-formulation and new techniques for solving DAE, are implemented in the SIMPACK code. Its capabilities are illustrated by simulation of multibody robot models.

Schwertassek, R.↗

DOE OSTI2025

Coupling flux balance analysis with reactive transport modeling through machine learning for rapid and stable simulation of microbial metabolic switching

Integrating genome-scale metabolic networks with reactive transport models (RTMs) provides a detailed description of the dynamic changes in microbial growth and metabolism. Despite promising demonstrations in the past, computational inefficiency has been pointed out as a critical issue to overcome because it requires repeated application of linear programming (LP) to obtain flux balance analysis (FBA) solutions in every time step and spatial grid. To address this challenge, we propose a new simulation method where we train and validate artificial neural networks (ANNs) using randomly sampled FBA solutions and incorporate the resulting surrogate FBA model (represented as algebraic equations) into RTMs as source/sink terms. We demonstrate the efficiency of our method via a case study of Shewanella oneidensis MR-1. During aerobic growth on lactate, S. oneidensis produces metabolic byproducts (such as pyruvate and acetate), which are subsequently consumed as alternative carbon sources when the preferred nutrients are depleted. To effectively simulate these complex dynamics, we used a cybernetic approach that models metabolic switches as the outcome of dynamic competition among multiple growth options. In both zero-dimensional batch and one-dimensional column configurations, the ANN-based surrogate models achieved substantial reduction of computational time by several orders of magnitude compared to the original LP-based FBA models. Moreover, the ANN models produced robust solutions without any special measures to prevent numerical instability. These developments significantly promote our ability to utilize genome-scale networks in complex, multi-physics, and multi-dimensional ecosystem modeling.

59 BASIC BIOLOGICAL SCIENCES↗

NASA NTRS1965

Matrix methods for calculating zeros, coefficients, Christoffel numbers, and derivatives of some orthogonal polynomials

Jacobi matrix method for calculating zeros, coefficients, Christoffel numbers, and derivatives of orthogonal polynomials

MATRIX ALGEBRA↗

NASA NTRS1992

Optimal matrix approximants in structural identification

Problems of model correlation and system identification are central in the design, analysis, and control of large space structures. Of the numerous methods that have been proposed, many are based on finding minimal adjustments to a model matrix sufficient to introduce some desirable quality into that matrix. In this work, several of these methods are reviewed, placed in a modern framework, and linked to other previously known ideas in computational linear algebra and optimization. This new framework provides a point of departure for a number of new methods which are introduced here. Significant among these is a method for stiffness matrix adjustment which preserves the sparsity pattern of an original matrix, requires comparatively modest computational resources, and allows robust handling of noisy modal data. Numerical examples are included to illustrate the methods presented herein.

Beattie, C. A.↗

DOE OSTI2026

Parametric reduced order models for graded lattice structures

Graded lattice structures, characterized by smoothly varying mechanical properties, hold significant promise for optimizing material distribution in advanced engineering applications. However, accurately modeling these structures poses substantial computational challenges due to the continuous geometric variations within their unit cells. Here, to address these challenges, this paper introduces a novel Efficient Reduced Order Model (EROM) that integrates the Matrix Discrete Empirical Interpolation Method (MDEIM) and Discrete Empirical Interpolation Method (DEIM) with polynomial regression to efficiently manage geometric parametrization in lattice structures. Unlike traditional reduced order models (ROMs) that require extensive precomputed libraries for each geometric configuration, our approach enables continuous geometric variations through a flexible algebraic formulation, significantly reducing computational costs while preserving high accuracy. The method constructs projection matrices for individual unit cells that can be efficiently assembled into global systems, leveraging the repetitive nature of lattice structures. Numerical studies demonstrate that our EROM achieves displacement errors below 1% and von Mises stress prediction errors below 4%, coupled with computational speedups exceeding two orders of magnitude compared to full-order simulations. The proposed method's modularity and scalability make it particularly suitable for design optimization and real-time simulation of functionally graded lattice structures, with applications spanning aerospace to biomedical engineering.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS1991

Observer-based robust-H-infinity control laws for uncertain linear systems

Based on the algebraic Riccati equation approach, this paper presents a simple and flexible method for designing observer-based robust-H-infinity control laws for linear systems with structured parameter uncertainty. The observer-based robust-H-infinity output-feedback control law, obtained by solving three augmented algebraic Riccati equations, provides both robust stability and disturbance attenuation with H-infinity-norm bound for the closed-loop uncertain linear system. Several tuning parameters are embedded into the augmented algebraic Riccati equations so that flexibility in finding the symmetric positive-definite solutions (and hence, the robust-H-infinity control laws) is significantly increased. A benchmark problem associated with a mass-spring system, which approximates the dynamics of a flexible structure, is used to illustrate the design methodologies, and simulation results are presented.

Shieh, Leang S.↗