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At least 325 records · Page 18

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Efficient Preconditioning of a High-Order Solver for Multiple Physics

This work addresses preconditioning approaches for an implicit high-order solver frame-work applied to multiple physics. The solver is based on a space-time spectral element method and matrix-free Newton-Krylov solver developed at NASA over the recent years. Within this context, most preconditioning methods are impractical, as the computational time and memory requirements scale poorly with increasing polynomial orders. To improve computational efficiency, we first describe a novel entity-based Block Jacobi preconditioner for the continuous-Galerkin solution of the linear-elasticity and linear-shell equations. Second, we introduce a multigrid algorithm to further reduce time-to-solution on stiff cases arising from continuous-and discontinuous-Galerkin discretizations. Results obtained on relevant single-physics reference solutions, demonstrate the feasibility of the methods, paving the way for high-order solutions of fully coupled multi-physics problems.

STMD↗

DOE OSTI2026

Statistical White-Line Analysis in High-Throughput TXM-XANES for Chemical State Quantification

The transmission X-ray microscopy (TXM) based X-ray absorption near-edge structure (XANES) technique provides three-dimensional mapping of element-specific chemical states at nanometer-scale spatial resolution and micrometer-scale fields of view. However, compared to conventional volume-averaged XANES (VA-XANES) measurements, the inherently small voxel size in TXM-XANES leads to a lower signal-to-noise ratio, making full-spectrum analysis computationally demanding and less robust. Here, we present the structural and compositional conditions for a statistical white-line analysis framework under which chemical state information can be directly extracted from the white-line peak position in voxel spectra without the need for voxel-wise background subtraction or normalization, under well-defined structural and compositional conditions. The method is validated on layered oxide cathode materials, where low-order polynomial fitting accurately reproduces white-line features, and the extracted energy distributions correlate strongly with VA-XANES results. This statistical approach enables high-throughput, dose-efficient, and noise-robust chemical state quantification in TXM-XANES, offering broad applicability to functional materials requiring nanoscale oxidation-state mapping.

TXM↗

DOE OSTI2024

Quadrature Based Neural Network Learning of Stochastic Hamiltonian Systems

Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs’ solutions, where we denoise the loss-by-moment calculations of the solutions. The integral pattern of the models transforms the source of the essential learning error from the discrepancy between the modified Hamiltonian and the true Hamiltonian in the classical HNN models into that between the integrals and their quadrature approximations. This transforms the challenging task of deriving the relation between the modified and the true Hamiltonians from the (stochastic) Hamilton–Jacobi PDEs, into the one that only requires invoking results from the numerical quadrature theory. Meanwhile, denoising via moments calculations gives a simpler data fitting method than, e.g., via probability density fitting, which may imply better generalization ability in certain circumstances. Numerical experiments validate the proposed learning strategy on several concrete Hamiltonian systems. The experimental results show that both the learned Hamiltonian function and the predicted solution of our quadrature-based model are more accurate than that of the corrected symplectic HNN method on a harmonic oscillator, and the three-point Gaussian quadrature-based model produces higher accuracy in long-time prediction than the Kramers–Moyal method and the numerics-informed likelihood method on the stochastic Kubo oscillator as well as other two stochastic systems with non-polynomial Hamiltonian functions. Moreover, the Hamiltonian learning error εH arising from the Gaussian quadrature-based model is lower than that from Simpson’s quadrature-based model. These demonstrate the superiority of our approach in learning accuracy and long-time prediction ability compared to certain existing methods and exhibit its potential to improve learning accuracy via applying precise quadrature formulae.

Mathematics↗

NASA NTRS1993

Positivity-preserving numerical schemes for multidimensional advection

This report describes the construction of an explicit, single time-step, conservative, finite-volume method for multidimensional advective flow, based on a uniformly third-order polynomial interpolation algorithm (UTOPIA). Particular attention is paid to the problem of flow-to-grid angle-dependent, anisotropic distortion typical of one-dimensional schemes used component-wise. The third-order multidimensional scheme automatically includes certain cross-difference terms that guarantee good isotropy (and stability). However, above first-order, polynomial-based advection schemes do not preserve positivity (the multidimensional analogue of monotonicity). For this reason, a multidimensional generalization of the first author's universal flux-limiter is sought. This is a very challenging problem. A simple flux-limiter can be found; but this introduces strong anisotropic distortion. A more sophisticated technique, limiting part of the flux and then restoring the isotropy-maintaining cross-terms afterwards, gives more satisfactory results. Test cases are confined to two dimensions; three-dimensional extensions are briefly discussed.

Leonard, B. P.↗

NASA NTRS1994

Spectral Methods for Computational Fluid Dynamics

As a tool for large-scale computations in fluid dynamics, spectral methods were prophesized in 1944, born in 1954, virtually buried in the mid-1960's, resurrected in 1969, evangalized in the 1970's, and catholicized in the 1980's. The use of spectral methods for meteorological problems was proposed by Blinova in 1944 and the first numerical computations were conducted by Silberman (1954). By the early 1960's computers had achieved sufficient power to permit calculations with hundreds of degrees of freedom. For problems of this size the traditional way of computing the nonlinear terms in spectral methods was expensive compared with finite-difference methods. Consequently, spectral methods fell out of favor. The expense of computing nonlinear terms remained a severe drawback until Orszag (1969) and Eliasen, Machenauer, and Rasmussen (1970) developed the transform methods that still form the backbone of many large-scale spectral computations. The original proselytes of spectral methods were meteorologists involved in global weather modeling and fluid dynamicists investigating isotropic turbulence. The converts who were inspired by the successes of these pioneers remained, for the most part, confined to these and closely related fields throughout the 1970's. During that decade spectral methods appeared to be well-suited only for problems governed by ordinary diSerential eqllations or by partial differential equations with periodic boundary conditions. And, of course, the solution itself needed to be smooth. Some of the obstacles to wider application of spectral methods were: (1) poor resolution of discontinuous solutions; (2) inefficient implementation of implicit methods; and (3) drastic geometric constraints. All of these barriers have undergone some erosion during the 1980's, particularly the latter two. As a result, the applicability and appeal of spectral methods for computational fluid dynamics has broadened considerably. The motivation for the use of spectral methods in numerical calculations stems from the attractive approximation properties of orthogonal polynomial expansions.

Zang, T. A.↗

NASA NTRS1996

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization

Fast methods are proposed for solving the system K(sub N)x = b resulting from the discretization of self-adjoint elliptic equations in three dimensional domains by the spectral element method. The domain is decomposed into hexahedral elements, and in each of these elements the discretization space is formed by polynomials of degree N in each variable. Gauss-Lobatto-Legendre (GLL) quadrature rules replace the integrals in the Galerkin formulation. This system is solved by the preconditioned conjugate gradients method. The conforming finite element space on the GLL mesh consisting of piecewise Q(sub 1) elements produces a stiffness matrix K(sub h) that is spectrally equivalent to the spectral element stiffness matrix K(sub N). The action of the inverse of K(sub h) is expensive for large problems, and is therefore replaced by a Schwarz preconditioner B(sub h) of this finite element stiffness matrix. The preconditioned operator then becomes B(sub h)(exp -l)K(sub N). The technical difficulties stem from the nonregularity of the mesh. Tools to estimate the convergence of a large class of new iterative substructuring and overlapping Schwarz preconditioners are developed. This technique also provides a new analysis for an iterative substructuring method proposed by Pavarino and Widlund for the spectral element discretization.

Casarin, Mario↗

NASA NTRS1999

How to Estimate Attitude from Vector Observations

The most robust estimators minimizing Wahba's loss function are Davenport's q method and the Singular Value Decomposition (SVD) method. The q method is faster than the SVD method with three or more measurements. The other algorithms are less robust since they solve the characteristic polynomial equation to find the maximum eigenvalue of Davenport's K matrix. They are only preferable when speed or processor power is an important consideration. Of these, Fast Optimal Attitude Matrix (FOAM) is the most robust and faster than the q method. Robustness is only an issue for measurements with widely differing accuracies, so the fastest algorithms, Quaternion ESTimator (QUEST), EStimator of the Optimal Quaternion (ESOQ), and ESOQ2, are well suited to star sensor applications.

Markley, F. Landis↗

NASA NTRS1998

Comparison of Response Surface and Kriging Models in the Multidisciplinary Design of an Aerospike Nozzle

The use of response surface models and kriging models are compared for approximating non-random, deterministic computer analyses. After discussing the traditional response surface approach for constructing polynomial models for approximation, kriging is presented as an alternative statistical-based approximation method for the design and analysis of computer experiments. Both approximation methods are applied to the multidisciplinary design and analysis of an aerospike nozzle which consists of a computational fluid dynamics model and a finite element analysis model. Error analysis of the response surface and kriging models is performed along with a graphical comparison of the approximations. Four optimization problems are formulated and solved using both approximation models. While neither approximation technique consistently outperforms the other in this example, the kriging models using only a constant for the underlying global model and a Gaussian correlation function perform as well as the second order polynomial response surface models.

Simpson, Timothy W.↗

NASA NTRS2011

Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions

This paper proposes an uncertainty analysis framework based on the characterization of the uncertain parameter space. This characterization enables the identification of worst-case uncertainty combinations and the approximation of the failure and safe domains with a high level of accuracy. Because these approximations are comprised of subsets of readily computable probability, they enable the calculation of arbitrarily tight upper and lower bounds to the failure probability. A Bernstein expansion approach is used to size hyper-rectangular subsets while a sum of squares programming approach is used to size quasi-ellipsoidal subsets. These methods are applicable to requirement functions whose functional dependency on the uncertainty is a known polynomial. Some of the most prominent features of the methodology are the substantial desensitization of the calculations from the uncertainty model assumed (i.e., the probability distribution describing the uncertainty) as well as the accommodation for changes in such a model with a practically insignificant amount of computational effort.

Crespo, Luis G.↗

NASA NTRS2007

Propagation of Computational Uncertainty Using the Modern Design of Experiments

This paper describes the use of formally designed experiments to aid in the error analysis of a computational experiment. A method is described by which the underlying code is approximated with relatively low-order polynomial graduating functions represented by truncated Taylor series approximations to the true underlying response function. A resource-minimal approach is outlined by which such graduating functions can be estimated from a minimum number of case runs of the underlying computational code. Certain practical considerations are discussed, including ways and means of coping with high-order response functions. The distributional properties of prediction residuals are presented and discussed. A practical method is presented for quantifying that component of the prediction uncertainty of a computational code that can be attributed to imperfect knowledge of independent variable levels. This method is illustrated with a recent assessment of uncertainty in computational estimates of Space Shuttle thermal and structural reentry loads attributable to ice and foam debris impact on ascent.

DeLoach, Richard↗

NASA NTRS1990

Analysis of spectral projectors in one-dimensional domains

A class of projection operators with values in a subspace of polynomials is analyzed. These projection operators are related to the Hilbert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthoronality relation satisfied by the Chebyshev polynomials. These results are used to study the approximation of a model fourth-order problem.

Maday, Y.↗

NASA NTRS1978

A Mach line panel method for computing the linearized supersonic flow over planar wings

A method is described for solving the linearized supersonic flow over planar wings using panels bounded by two families of Mach lines. Polynomial distributions of source and doublet strength lead to simple, closed form solutions for the aerodynamic influence coefficients, and a nearly triangular matrix yields rapid solutions for the singularity parameters. The source method was found to be accurate and stable both for analysis and design boundary conditions. Similar results were obtained with the doublet method for analysis boundary conditions on the portion of the wing downstream of the supersonic leading edge, but instabilities in the solution occurred for the region containing a portion of the subsonic leading edge. Research on the method was discontinued before this difficulty was resolved.

Ehlers, F. E.↗

NASA NTRS1994

Coupling equivalent plate and finite element formulations in multiple-method structural analyses

A coupled multiple-method analysis procedure for use late in conceptual design or early in preliminary design of aircraft structures is described. Using this method, aircraft wing structures are represented with equivalent plate models, and structural details such as engine/pylon structure, landing gear, or a 'stick' model of a fuselage are represented with beam finite element models. These two analysis methods are implemented in an integrated multiple-method formulation that involves the assembly and solution of a combined set of linear equations. The corresponding solution vector contains coefficients of the polynomials that describe the deflection of the wing and also the components of translations and rotations at the joints of the beam members. Two alternative approaches for coupling the methods are investigated; one using transition finite elements and the other using Lagrange multipliers. The coupled formulation is applied to the static analysis and vibration analysis of a conceptual design model of a fighter aircraft. The results from the coupled method are compared with corresponding results from an analysis in which the entire model is composed of finite elements.

VIBRATION ANALYSIS↗

DOE OSTI2025

Polynomial Scaling Localized Active Space Unitary Selective Coupled Cluster Singles and Doubles

We present a polynomial-scaling algorithm for the localized active space unitary selective coupled cluster singles and doubles (LAS-USCCSD) method. In this approach, cluster excitations are selected based on a threshold ϵ determined by the absolute gradients of the LAS-UCCSD energy with respect to cluster amplitudes. Using the generalized Wick’s theorem for multireference wave functions, we derive the gradient expression as a polynomial function of one-, two-, and three-body reduced density matrices and 1- and 2-electron integrals, valid for any multireference wave function. The resulting gradient implementation exhibits a memory scaling of 𝒪(N 6 ), with N spin orbitals in the combined active space of all fragments. The variational quantum eigensolver is used to optimize the selected cluster excitations on a quantum simulator. Furthermore, by plotting the energy error, defined as the difference between the LAS-USCCSD and corresponding CASCI energies, against the inverse cluster amplitude selection threshold (ϵ –1 ) for polyene chains containing 2 to 5 π-bond units, we establish a relationship between the energy error and the threshold. To further validate the accuracy of LAS-USCCSD, we computed the cis–trans isomerization energy of stilbene (a 20-qubit system) and the magnetic coupling constant of the tris-hydroxo-bridged chromium dimer [Cr 2 (OH) 3 (NH 3 ) 6 ] 3+ (evaluated as both 12- and 20-qubit systems) using the Qiskit-Qulacs simulator. Assessing such examples is important to determine the practical feasibility of quantum simulations for chemically realistic systems. Toward this goal, with the LAS-USCCSD algorithm we estimated the quantum resources required for simulating an active space of (30e,22o) in [Cr 2 (OH) 3 (NH 3 ) 6 ] 3+ , a size that remains beyond the reach of current quantum simulators for accurate treatment.

Algorithms↗

NASA NTRS1975

Analysis of multiloop, multirate sampled-data systems

Based upon new identities between z-transforms at a basis rate, z-transforms at faster rates, and modified z-transforms, the equivalence between the frequency decomposition method and the switch decomposition method is precisely presented so that results of one method are easily obtained from results of the other. Next, a method is developed for determining the closed loop transfer function of multiloop, multirate sampled-data systems with noninteger ratio sampling rates. Previously, this process involved solving a complex system of equations with rational polynomial coefficients. Herein, this is avoided by introducing a systematic decomposition of matrix operators which naturally arise from the switch decomposition method. The matrix operators are simplified by introducing the shifted transforms of signals sampled at one of the faster rates.

Boykin, W. H.↗

NASA NTRS1989

On the method of matched asymptotic expansions

The present evaluation of the method of asymptotic expansions (MAE) indicates that the various terms of the common solution of MAE can be generated as polynomials in stretched variables, without actually solving them from the outer solution, as is currently the practice. It is also noted that the common solution of the MAE and the intermediate solution of the singular-perturbation method are the same; these methods therefore yield identical results for a certain class of problems. Two illustrative problems are treated.

Naidu, D. S.↗

DOE OSTI2026

Fermionic mean-field dynamics for spin systems beyond free fermions

We introduce the fermionized time-dependent Hartree–Fock (fTDHF), a real-time quantum dynamics method for spin-1/2 Hamiltonians following their mapping to fermions via the Jordan-Wigner transformation. fTDHF is formally equivalent to exact dynamics in the case of free fermions, and can efficiently handle non-local string operators arising from long-range interactions via transition matrix elements between non-orthogonal Slater determinants. We show that the fTDHF method can be implemented on a classical computer with a cost that scales polynomially with system size, and linearly with the time steps. We benchmark fTDHF against exact dynamics on three separate spin-1/2 models, representing adiabatic preparation of states with long-range correlations, disorder-driven observation of many-body localization, and particle production in the Schwinger model. For each of these systems, fTDHF is shown to reproduce the qualitative dynamics generated by the exact evolutions, while maintaining a simple physical picture due to its mean-field nature.

Dutta, Rishab↗

NASA NTRS1985

Lubrication of nonconformal contacts

Minimum film thickness results for piezoviscous-rigid regime of lubrication are developed for a compressible Newtonian fluid with Roelands viscosity. The results provide a basis for the analysis and design of a wide range of machine elements operating in the piezoviscous-rigid regime of lubrication. A new numerical method of calculating elastic deformation in contact stresses is developed using a biquadratic polynomial to approximate the pressure distribution on the whole domain analyzed. The deformation of every node is expressed as a linear combination of the nodal pressures whose coefficients can be combined into an influence coefficient matrix. This approach has the advantages of improved numerical accuracy, less computing time and smaller storage size required for influence matrix. The ideal elastohydrodynamic lubrication is extended to real bearing systems in order to gain an understanding of failure mechanisms in machine elements. The improved elastic deformation calculation is successfully incorporated into the EHL numerical scheme. Using this revised numerical technique and the flow factor model developed by Patir and Cheng (1978) the surface roughness effects on the elastohydrodynamic lubrication of point contact is considered. Conditions typical of an EHL contact in the piezoviscous-elastic regime entrained in pure rolling are investigated. Results are compared with the smooth surface solutions. Experiments are conducted to study the transient EHL effects in instrument ball bearings.

Jeng, Y. R.↗