SEARCH · Search NASA

Results for “Computer Algebra”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 865 records · Page 48

All sources NASA NTRS DOE OSTI

DOE OSTI2024

A cluster of results on amplituhedron tiles

Abstract The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in $$\mathcal {N}=4$$ N = 4 super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the $$m=4$$ m = 4 amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for $$\text{ Gr}_{4,n}$$ Gr 4 , n . Secondly, we exhibit a tiling of the $$m=4$$ m = 4 amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for $$\text{ Gr}_{4,n}$$ Gr 4 , n . This paper is a companion to our previous paper “Cluster algebras and tilings for the $$m=4$$ m = 4 amplituhedron.”

Physics↗

NASA NTRS1995

Application of Navier-Stokes code PAB3D with kappa-epsilon turbulence model to attached and separated flows

A three-dimensional Navier-Stokes solver was used to determine how accurately computations can predict local and average skin friction coefficients for attached and separated flows for simple experimental geometries. Algebraic and transport equation closures were used to model turbulence. To simulate anisotropic turbulence, the standard two-equation turbulence model was modified by adding nonlinear terms. The effects of both grid density and the turbulence model on the computed flow fields were also investigated and compared with available experimental data for subsonic and supersonic free-stream conditions.

Abdol-Hamid, Khaled S.↗

NASA NTRS1978

ORACLS: A system for linear-quadratic-Gaussian control law design

A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.

Armstrong, E. S.↗

NASA NTRS2005

Toward Question-Asking Machines: The Logic of Questions and the Inquiry Calculus

For over a century, the study of logic has focused on the algebra of logical statements. This work, first performed by George Boole, has led to the development of modern computers, and was shown by Richard T. Cox to be the foundation of Bayesian inference. Meanwhile the logic of questions has been much neglected. For our computing machines to be truly intelligent, they need to be able to ask relevant questions. In this paper I will show how the Boolean lattice of logical statements gives rise to the free distributive lattice of questions thus defining their algebra. Furthermore, there exists a quantity analogous to probability, called relevance, which quantifies the degree to which one question answers another. I will show that relevance is not only a natural generalization of information theory, but also forms its foundation.

Knuth,Kevin H.↗

DOE OSTI2017

Operator bases, S-matrices, and their partition functions

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. Here in this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S- matrix in the form of soft limits. The most naÏve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations. Finally, although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2019

Pressio

Reduced-order Models (ROMs). ROMs are "compressed" approximations of high-fidelity computational models. Implementing ROM methods in production codes is typically intrusive, i.e. the native codes must be adapted and modified, with individual ROM methods reimplemented in every application code. This clearly constitutes an obstacle to users who want to use ROMs. To address this, we have developed Pressio, which is a C++ header-only ROM library that can be used by an application code in a non-intrusive fashion. The main objectives behind the design of Pressio are: (a) support for general computational models (data structures, PDE types); (b) heterogeneous hardware (CPU, GPU); (c) performance; and (d) extensibility. Using metaprogramming techniques, Pressio detects and leverages the application?s native linear algebra data structures (e.g. vector and matrix) to run the ROM methods. The application only needs to expose the velocity vector and Jacobian matrix via an adapter class. Pressio checks at compile time that the target application meets the correct API. The library is structured in several modular packages including (a) non-linear least-square solvers (e.g. Gauss-Newton with and without line-search); (b) singular value decomposition (for the computation of the ROM generalized coordinates) and (c) algebraic hyper-reduction.SAND2019-5878 M

Blonigan, Patrick↗

DOE OSTI2019

Pressio v.1.0

Blonigan, Patrick↗

NASA NTRS1973

Computer checking of rotational line intensity factors for diatomic transitions.

Previously published analytical formulae for the rotational line intensity factors of diatomic molecules have been compared with results from a comprehensive computer program, which determines numerical intensity factors for both spin-allowed and spin-forbidden electric and magnetic dipole transitions in diatomic molecules. The comparison uncovered several typographical errors and a few algebraic errors in the published formulae. The changes required in the formulae to give agreement with the results from the computer program are tabulated.

Whiting, E. E.↗

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↗

DOE OSTI2022

Nucleon-pair truncation of the shell model for medium-heavy nuclei

Background: Establishing computationally tractable models of atomic nuclei is a long-time goal of nuclear structure physics. A flexible framework which easily includes excited states and many-body correlations is the configuration-interaction shell model (SM), but the exponential growth of the basis means one needs an efficient truncation scheme, ideally one that includes both deformation and pairing correlations. Purpose: Here, we propose an efficient truncation scheme of the SM: starting from a pair condensate variationally defined by Hartree-Fock single-particle states and the particle-number conserved Bardeen-Cooper-Schrieffer (NBCS) approximation, we carry out projection of states with good angular momentum. Methods: After generating Hartree-Fock single-particle states with Kramers degeneracy in a SM space, we optimize the pair amplitudes in the NBCS by minimizing the energy, and then use linear algebra projection (LAP) of states with good angular momentum. Both NBCS and LAP are computationally fast. Results: Our calculations yield good agreement with full configuration-interaction SM calculations for low-lying states of transitional and rotational nuclei with axially symmetric and triaxial deformation in medium- and heavy-mass regions: 44,46,48 Ti, 48,50 Cr, 52 Fe, 60,62,64 Zn, 66,68 Ge, 68 Se, and 108,110 Xe. We predict low-lying states of 112–114 Ba and 116–120 Ce, nuclei difficult to reach by large-scale SM calculations. Conclusions: Both pair correlation and the configuration mixing between different intrinsic states play a key role in reproducing collectivity and shape coexistence, demonstrating the utility of this truncation scheme of the SM to study transitional and deformed nuclei.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1993

Some practical turbulence modeling options for Reynolds-averaged full Navier-Stokes calculations of three-dimensional flows

New turbulence modeling options recently implemented for the 3D version of Proteus, a Reynolds-averaged compressible Navier-Stokes code, are described. The implemented turbulence models include: the Baldwin-Lomax algebraic model, the Baldwin-Barth one-equation model, the Chien k-epsilon model, and the Launder-Sharma k-epsilon model. Features of this turbulence modeling package include: well documented and easy to use turbulence modeling options, uniform integration of turbulence models from different classes, automatic initialization of turbulence variables for calculations using one- or two-equation turbulence models, multiple solid boundaries treatment, and fully vectorized L-U solver for one- and two-equation models. Good agreements are obtained between the computational results and experimental data. Sensitivity of the compressible turbulent solutions with the method of y(+) computation, the turbulent length scale correction, and some compressibility corrections are examined in detail. Test cases show that the highly optimized one- and two-equation turbulence models can be used in routine 3D Navier-Stokes computations with no significant increase in CPU time as compared with the Baldwin-Lomax algebraic model.

Bui, Trong T.↗

DOE OSTI2024

Integrating subsystem embedding subalgebras and coupled cluster Green’s function: a theoretical foundation for quantum embedding in excitation manifold

Here, in this study, we introduce a novel approach to coupled-cluster Green's function (CCGF) embedding by seamlessly integrating conventional CCGF theory with the state-of-the-art sub-system embedding sub-algebras coupled cluster (SES-CC) formalism. This integration focuses primarily on delineating the characteristics of the sub-system and the corresponding segments of the Green's function, defined explicitly by active orbitals. Crucially, our work involves the adaptation of the SES-CC paradigm, addressing the left eigenvalue problem through a distinct form of Hamiltonian similarity transformation. This advancement not only facilitates a comprehensive representation of the interaction between the embedded sub-system and its surrounding environment but also paves the way for the quantum mechanical description of multiple embedded domains, particularly by employing the emergent quantum flow algorithms. Our theoretical underpinnings further set the stage for a generalization to multiple embedded sub-systems. This expansion holds significant promise for the exploration and application of non-equilibrium quantum systems, enhancing the understanding of system–environment interactions. In doing so, the research underscores the potential of SES-CC embedding within the realm of quantum computations and multi-scale simulations, promising a good balance between accuracy and computational efficiency.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1984

Theoretical analysis of aircraft afterbody flow

The strongly interactive flow field about aircraft afterbodies is investigated using computational techniques by which the thin-shear-layer formulation of the compressible, Reynolds-averaged Navier-Stokes equations is solved. A time-dependent implicit numerical algorithm is used to obtain solutions for a variety of afterbody and nozzle geometries, within the class of bodies of revolution, for both subsonic and supersonic external flow, and for sonic and supersonic underexpanded jets. Only centered nozzles with either a sharp lip or a blunt base are considered. In all cases, computed results are compared with experimental data taken at flight Reynolds numbers for like-flow conditions. Turbulence closure is realized using algebraic eddy-viscosity concepts. A new and unique adaptive-grid technique is used to resolve flow regimes with large gradients and to improve the accuracy and efficiency of the computational scheme. Special singular point boundary conditions are used for similar purposes, and are especially effective for highly under-expanded jets. For all cases considered, except one with a very large base-to-nozzle-exit-diameter ratio, the agreement with experimental measurements is excellent. For geometries with large base regions, enhancements in the turbulence transport model are necessary to support improvements in the flow-field simulation.

Deiwert, G. S.↗

DOE OSTI2019

A Shift Selection Strategy for Parallel Shift-Invert Spectrum Slicing in Symmetric Self-Consistent Eigenvalue Computation

The central importance of large scale eigenvalue problems in scientific computation necessitates the development massively parallel algorithms for their solution. Recent advances in dense numerical linear algebra have enabled the routine treatment of eigenvalue problems with dimensions on the order of hundreds of thousands on the world's largest supercomputers. In cases where dense treatments are not feasible, Krylov subspace methods offer an attractive alternative due to the fact that they do not require storage of the problem matrices. However, demonstration of scalability of either of these classes of eigenvalue algorithms on computing architectures capable of expressing excessive parallelism is non-trivial due to communication requirements and serial bottlenecks, respectively. In this work, we introduce the SISLICE method: a parallel shift-invert algorithm for the solution of the symmetric self-consistent field (SCF) eigenvalue problem. The SISLICE method drastically reduces the communication requirement of current parallel shift-invert eigenvalue algorithms through various shift selection and migration techniques based on density of states estimation and k-means clustering, respectively. This work demonstrates the robustness and parallel performance of the SISLICE method on a representative set of SCF eigenvalue problems and outlines research directions which will be explored in future work.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1973

High speed flow past wings

The analytical solution to the transonic small perturbation equation which describes steady compressible flow past finite wings at subsonic speeds can be expressed as a nonlinear integral equation with the perturbation velocity potential as the unknown function. This known formulation is substituted by a system of nonlinear algebraic equations to which various methods are applicable for its solution. Due to the presence of mathematical discontinuities in the flow solutions, however, a main computational difficulty was to ensure uniqueness of the solutions when local velocities on the wing exceeded the speed of sound. For continuous solutions this was achieved by embedding the algebraic system in an one-parameter operator homotopy in order to apply the method of parametric differentiation. The solution to the initial system of equations appears then as a solution to a Cauchy problem where the initial condition is related to the accompanying incompressible flow solution. In using this technique, however, a continuous dependence of the solution development on the initial data is lost when the solution reaches the minimum bifurcation point. A steepest descent iteration technique was therefore, added to the computational scheme for the calculation of discontinuous flow solutions. Results for purely subsonic flows and supersonic flows with and without compression shocks are given and compared with other available theoretical solutions.

Norstrud, H.↗

DOE OSTI2016

Quantum transport modelling of silicon nanobeams using heterogeneous computing scheme

We report the development of a powerful method for quantum transport calculations of nanowire/nanobeam structures with large cross sectional area. Our approach to quantum transport is based on Green's functions and tight-binding potentials. A linear algebraic formulation allows us to harness the massively parallel nature of Graphics Processing Units (GPUs) and our implementation is based on a heterogeneous parallel computing scheme with traditional processors and GPUs working together. Using our software tool, the electronic and quantum transport properties of silicon nanobeams with a realistic cross sectional area of ∼22.7 nm{sup 2} and a length of ∼81.5 nm—comprising 105 000 Si atoms and 24 000 passivating H atoms in the scattering region—are investigated. The method also allows us to perform significant averaging over impurity configurations—all possible configurations were considered in the case of single impurities. Finally, the effect of the position and number of vacancy defects on the transport properties was considered. It is found that the configurations with the vacancies lying closer to the local density of states (LDOS) maxima have lower transmission functions than the configurations with the vacancies located at LDOS minima or far away from LDOS maxima, suggesting both a qualitative method to tune or estimate optimal impurity configurations as well as a physical picture that accounts for device variability. Finally, we provide performance benchmarks for structures as large as ∼42.5 nm{sup 2} cross section and ∼81.5 nm length.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Evaporative thermal resistance and its influence on microlayer evaporation

The ability to predict various aspects of boiling is important for both safety and design of water-cooled reactors. Component-scale modelling is necessarily done by semi-mechanistic, semi-empirical approaches, generally based on heat flux partitioning. This needs to be informed by multiple detailed experimental observations of boiling behavior. Such measurements are difficult to make, and much effort is being applied to microscopic computational modelling of the detailed boiling process. Aims of such activities include the mechanistic prediction of growth and detachment of one bubble during nucleate boiling, and the quantification of the associated heat transfer partitioning (evaporation, micro-convection, quenching). We here study a subset of the full problem, and focus on the thermal response of the boiling substrate to the early growth of a single vapour bubble during pool boiling of water at atmospheric pressure. In such conditions, a thin liquid film ('microlayer') is left behind beneath the bubble as this expands. The microlayer evaporates due to the temperature difference between the solid-liquid and vapour-liquid interface. Its thickness being small, of order of microns, the local evaporation rate per unit area of bubble base can be high, up to one kilogram per second. Microlayer evaporation can therefore be important in early stages of bubble growth. In this contribution, thermal response of the boiling substrate to microlayer evaporation is studied with a two-dimensional, axisymmetric transient conduction model, coupled with an algebraic sub-model of the liquid film. The flow of heat through the microlayer is modelled as conduction in the liquid and with a pseudo-convective 'evaporative' heat transfer coefficient at the liquid-vapor interface, accounting for the interface superheat driving the evaporation process. Thermal coupling (conjugate heat transfer) between microlayer and solid substrate is obtained with a time dependent non-homogeneous Neumann boundary condition, derived from the algebraic microlayer sub-model and applied at the top surface of the solid computational domain. The thermal resistance associated to the evaporative heat transfer mechanism is shown to depend on a highly uncertain parameter, termed an 'evaporation coefficient'. After some exploratory calculations of the likely importance of the evaporative resistance in the present context, the model is employed to interpret some recent microlayer measurements. Satisfactory agreement between computed and measured evolution of the microlayer profile is obtained if small values (between 0.01 and 0.10) of the evaporation coefficient are used. The order of magnitude of the predicted evaporation rate from the microlayer is consistent with experimental observations. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2018

Cyclotomic Shuffles

Analogues of 1-shuffle elements for complex reflection groups of type G(m,1,n) are introduced. A geometric interpretation for G(m,1,n) in terms of rotational permutations of polygonal cards is given. We compute the eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra of the group G(m,1,n). Considering shuffling as a random walk on the group G(m,1,n), we estimate the rate of convergence to randomness of the corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue in the cyclotomic Hecke algebra H(m,1,n) for m=2 and small n.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗