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At least 793 records · Page 44

Adaptive Meshing Techniques for Viscous Flow Calculations on Mixed Element Unstructured Meshes

An adaptive refinement strategy based on hierarchical element subdivision is formulated and implemented for meshes containing arbitrary mixtures of tetrahendra, hexahendra, prisms and pyramids. Special attention is given to keeping memory overheads as low as possible. This procedure is coupled with an algebraic multigrid flow solver which operates on mixed-element meshes. Inviscid flows as well as viscous flows are computed an adaptively refined tetrahedral, hexahedral, and hybrid meshes. The efficiency of the method is demonstrated by generating an adapted hexahedral mesh containing 3 million vertices on a relatively inexpensive workstation.

Mavriplis, D. J.↗

Computational methods in nonlinear structural and solid mechanics; Proceedings of the Symposium, Washington, D.C., October 6-8, 1980. Symposium sponsored by the George Washington University and NASA

Topics discussed include computational strategies for nonlinear problems in structural mechanics, time integration techniques and the numerical solution of nonlinear algebraic equations, material characterization and nonlinear fracture mechanics, nonlinear interaction problems, and seismic response and the nonlinear analysis of concrete structures. Also considered are nonlinear problems for nuclear reactors, crash dynamics and impact problems, nonlinear problems of fibrous composites and advanced nonlinear applications, and computerized symbolic manipulation and nonlinear analysis software systems.

Noor, A. K.↗

Time-domain finite elements in optimal control with application to launch-vehicle guidance

A time-domain finite element method is developed for optimal control problems. The theory derived is general enough to handle a large class of problems including optimal control problems that are continuous in the states and controls, problems with discontinuities in the states and/or system equations, problems with control inequality constraints, problems with state inequality constraints, or problems involving any combination of the above. The theory is developed in such a way that no numerical quadrature is necessary regardless of the degree of nonlinearity in the equations. Also, the same shape functions may be employed for every problem because all strong boundary conditions are transformed into natural or weak boundary conditions. In addition, the resulting nonlinear algebraic equations are very sparse. Use of sparse matrix solvers allows for the rapid and accurate solution of very difficult optimization problems. The formulation is applied to launch-vehicle trajectory optimization problems, and results show that real-time optimal guidance is realizable with this method. Finally, a general problem solving environment is created for solving a large class of optimal control problems. The algorithm uses both FORTRAN and a symbolic computation program to solve problems with a minimum of user interaction. The use of symbolic computation eliminates the need for user-written subroutines which greatly reduces the setup time for solving problems.

Bless, Robert R.↗

Luneburg lens and optical matrix algebra research

Planar, as opposed to channelized, integrated optical circuits (IOCs) were stressed as the basis for computational devices. Both fully-parallel and systolic architectures are considered and the tradeoffs between the two device types are discussed. The Kalman filter approach is a most important computational method for many NASA problems. This approach to deriving a best-fit estimate for the state vector describing a large system leads to matrix sizes which are beyond the predicted capacities of planar IOCs. This problem is overcome by matrix partitioning, and several architectures for accomplishing this are described. The Luneburg lens work has involved development of lens design techniques, design of mask arrangements for producing lenses of desired shape, investigation of optical and chemical properties of arsenic trisulfide films, deposition of lenses both by thermal evaporation and by RF sputtering, optical testing of these lenses, modification of lens properties through ultraviolet irradiation, and comparison of measured lens properties with those expected from ray trace analyses.

Wood, V. E.↗

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↗

Dissipative magnetohydrodynamics for nonresistive relativistic plasmas: An implicit second-order flux-conservative formulation with stiff relaxation

Based on a 14-moment closure for nonresistive (general-) relativistic viscous plasmas, here we describe a new numerical scheme that is able to handle all first-order dissipative effects (heat conduction, bulk and shear viscosities), as well the anisotropies induced by the presence of magnetic fields. The latter is parametrized in terms of a thermal gyrofrequency or, equivalently, a thermal Larmor radius and allows to correctly capture the thermal Hall effect. By solving an extended Israel-Stewart-like system for the dissipative quantities that enforces algebraic constraints via stiff-relaxation, we are able to cast all first-order dissipative terms in flux-divergence form. This allows us to apply traditional high-resolution shock capturing methods to the equations, making the system suitable for the numerical study of highly turbulent flows. We present several numerical tests to assess the robustness of our numerical scheme in flat spacetime. The 14-moment closure can seamlessly interpolate between the highly collisional limit found in neutron star mergers, and the highly anisotropic limit of relativistic Braginskii magnetohydrodynamics appropriate for weakly collisional plasmas in black-hole accretion problems. We believe that this new formulation and numerical scheme will be useful for a broad class of relativistic magnetized flows.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo

In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well suited to incorporating into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the right-hand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g., multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine-scale random fields in a hierarchical fashion from coarse-scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in $L^2$ which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility.

algebraic multigrid↗

Three-dimensional inviscid analysis of the scramjet inlet flow field

A computer code has been developed to analyze the inviscid flow field in a supersonic combustion ramjet (scramjet) inlet. The code uses the three-dimensional Euler equations in full conservation form to describe the inlet flow. An algebraic numerical coordinate transformation is used to generate a set of boundary-fitted curvilinear coordinates. The governing equations are solved by a time-asymptotic, unsplit, two-step, finite-difference method. This method is highly efficient on the vector processing computers for which the current code is written. Detailed results are presented for two scramjet inlet configurations over a range of Mach numbers. The calculated results are compared with the available experimental and theoretical results.

Kumar, A.↗

The analysis of a rocket tomography measurement of the N2+3914A emission and N2 ionization rates in an auroral arc

Techniques were developed for recovering two-dimensional distributions of auroral volume emission rates from rocket photometer measurements made in a tomographic spin scan mode. These tomographic inversion procedures are based upon an algebraic reconstruction technique (ART) and utilize two different iterative relaxation techniques for solving the problems associated with noise in the observational data. One of the inversion algorithms is based upon a least squares method and the other on a maximum probability approach. The performance of the inversion algorithms, and the limitations of the rocket tomography technique, were critically assessed using various factors such as (1) statistical and non-statistical noise in the observational data, (2) rocket penetration of the auroral form, (3) background sources of emission, (4) smearing due to the photometer field of view, and (5) temporal variations in the auroral form. These tests show that the inversion procedures may be successfully applied to rocket observations made in medium intensity aurora with standard rocket photometer instruments. The inversion procedures have been used to recover two-dimensional distributions of auroral emission rates and ionization rates from an existing set of N2+3914A rocket photometer measurements which were made in a tomographic spin scan mode during the ARIES auroral campaign. The two-dimensional distributions of the 3914A volume emission rates recoverd from the inversion of the rocket data compare very well with the distributions that were inferred from ground-based measurements using triangulation-tomography techniques and the N2 ionization rates derived from the rocket tomography results are in very good agreement with the in situ particle measurements that were made during the flight. Three pre-prints describing the tomographic inversion techniques and the tomographic analysis of the ARIES rocket data are included as appendices.

Mcdade, Ian C.↗

Evaluation of an optimal guidance algorithm for aero-assisted orbit transfer

A detailed evaluation is performed of a guidance algorithm for aeroassisted orbit transfer that was developed earlier based on the method of matched asymptotic expansions. It is shown that, by exploiting the structure of the matched asymptotic expansion solution procedure, the original problem which requires the solution of a set of 20 implicit algebraic equations can be reduced to a problem of six implicit equations in six unknowns. The main contribution here is that it was possible to obtain a solution that is near optimal, requires a minimum of computation, and thus can be implemented in real time and on-board the vehicle. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the matched asymptotic expansion problem to obtain the feedback controls.

Melamed, N.↗

Two-Level Sketching Alternating Anderson Acceleration for Complex Physics Applications

We present a novel two-level sketching extension of the Alternating Anderson–Picard (AAP) method for accelerating fixed-point iterations in challenging single- and multiphysics simulations governed by discretized PDEs. Our approach combines a static, physics-based projection that reduces the least-squares (LS) problem to the most informative field (e.g., via Schur-complement insight) with a dynamic, algebraic sketching stage driven by a backward stability analysis under Lipschitz continuity. We introduce inexpensive estimators for stability thresholds and cache-aware randomized selection strategies to balance computational cost against memory access overhead. The resulting algorithm solves reduced LS systems in place, minimizes memory footprints, and seamlessly alternates between low-cost Picard updates and Anderson mixing. Implemented in Julia, our two-level sketching AAP achieves up to 50% time-to-solution reductions compared to standard Anderson acceleration—without degrading convergence rates—on benchmark problems including Stokes, 𝑝-Laplacian, bidomain, and Navier–Stokes formulations at varying problem sizes. These results demonstrate the method’s robustness, scalability, and potential for integration into high-performance scientific computing frameworks. Our implementation is available open source in the AAP.jl library.

Barnafi, Nicolas [University of Chile, Santiago]↗

X-ray Mapping of Terrestrial and Extraterrestrial Materials Using the Electron Microprobe

Lunar samples returned from the Apollo program motivated development of the Bence-Albee algorithm for the rapid and accurate analysis of lunar materials, and established interlaboratory comparability through its common use. In the analysis of mineral and rock fragments it became necessary to combine micro- and macroscopic analysis by coupling electron-probe microanalysis (EPMA) with automated stage point counting. A coarse grid that included several thousand points was used, and initially wavelength-dispersive (WDS) and later energydispersive (EDS) data were acquired at discrete stage points using approx. 5 sec count times. A approx 50 micrometer beam diameter was used for WDS and up to 500 micrometer beam diameter for EDS analysis. Average analyses of discretely sampled phases were coupled with the point count data to calculate the bulk composition using matrix algebra. Use of a defocused beam resulted in a contribution from multiple phases to each analytical point, and the analytical data were deconvolved relative to end-member phase chemistry on the fly. Impressive agreement was obtained between WDS and EDS measurements as well as comparison with bulk chemistry obtained by other methods. In the 30 years since these methods were developed, significant improvements in EPMA automation and computer processing have taken place. Digital beam control allows routine collection of x-ray maps by EDS, and stage mapping for WDS is conducted continuously at slew speed and incrementally by sampling at discrete points. Digital pulse processing in EDS systems has significantly increased the throughput for EDS mapping, and the ongoing development of Si-drift detector systems promises mapping capabilities rivaling WDS systems. Spectrum imaging allows a data cube of EDS spectra to be acquired and sophisticated processing of the original data is possible using matrix algebra techniques. The study of lunar and meteoritic materials includes the need to conveniently: (1) Characterize the sample at microscopic and macroscopic scales with relatively high sensitivity, (2) Determine the modal abundance of minerals, and (3) Identify and relocate discrete features of interest in terms of size and chemistry. The coupled substitution of cations in minerals can result in significant variation in mineral chemistry, but at similar average Z, leading to poor backscattered-electron (BSE) contrast discrimination of mineralogy. It is necessary to discriminate phase chemistry at both the trace element level and the major element level. To date, the WDS of microprobe systems is preferred for mapping due to high throughput and the ability to obtain the necessary intensity to discriminate phases at both trace and major element concentrations. It is desirable to produce fully quantitative compositional maps of geological materials, which requires the acquisition of k-ratio maps that are background and dead-time corrected, and which have been corrected by phi(delta z> or an equivalent algorithm at each pixel. To date, turnkey systems do not allow the acquisition of k-ratio maps and the rigorous correction in this manner. X-ray maps of a chondrule from the Ourique meteorite, and a comb-layered xenolith from the San Francisco volcanic field, have been analyzed and processed to extract phase information. The Ourique meteorite presents a challenge due to relatively low BSE contrast, and has been studied using spectrum imaging. X-ray maps for Si, Mg, and FeK(alpha) were used to produce RGB images. The xenolith sample contains sector-zoned augite, olivine, plagioclase, and basaltic glass. X-ray maps were processed using Lispix and ImageJ software to produce mineral phase maps. The x-ray maps for Mg, Ca, and Ti were used with traceback to generate binary images that were converted to RGB images. These approaches are successful in discriminating phases, but it is desirable to achieve the methods that were used on lunar samples 30 years ago on current microprobe systems. Curnt research includes x-ray mapping analysis of the Dalgety Downs chondrite by micro x-ray fluorescence and spectrum imaging, in collaboration with Kenny Witherspoon of IXRF Systems and Dale Newbury of NIST.

Carpenter, P.↗

Response of joint dominated space structures

An efficient linearization method is presented for calculating the transient response of nonlinear systems due to initial disturbances. The method is an extension of the describing function approach in which the steady state response of the system is calculated by representing the nonlinear element, typically joints in the case of space structures, by impedances which are functions of the amplitude of response. Thus, the problem of solving the differential equation for the steady state response becomes one of solving a set of nonlinear algebraic equations involving the steady state amplitudes and phases of the system. It is shown that for the transient case the steady state impedances can be averaged over the range of responses in order to provide equivalent values of stiffness and damping that, for a given set of initial displacements, may be treated as being constant for purposes of calculating system response. Single degree of freedom system are used to demonstrate the method and to develop an approach for optimizing the joint's characteristics so as to minimize transient response times. The use of this method for response estimation and optimization in multiple degree of freedom systems is investigated.

Source record↗

Machine Learning meets Algebraic Combinatorics: A Suite of Benchmark Datasets to Accelerate AI for Mathematics Research

The use of benchmark datasets has become an important engine of progress in machine learning (ML) over the past 15 years. Recently there has been growing interest in utilizing machine learning to drive advances in research-level mathematics. However, off-the-shelf solutions often fail to deliver the types of insights required by mathematicians. This suggests the need for new ML methods specifically designed with mathematics in mind. The question then is: what benchmarks should the community use to evaluate these? On the one hand, toy problems such as learning the multiplicative structure of small finite groups have become popular in the mechanistic interpretability community whose perspective on explainability aligns well with the needs of mathematicians. While toy datasets are a useful benchmark for initial work, they lack the scale, complexity, and sophistication of many of the principal objects of study in modern mathematics. To address this, we introduce a new collection of benchmark datasets, Algebraic Combinatorics Benchmarks (ACBench), representing either classic or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. After describing the datasets, we discuss the challenges involved in constructing “good” mathematics benchmarks, describe baseline model performance, and discuss some of the insights these datasets can provide that may be of interest even to those who are not interested in mathematics research itself.

97 MATHEMATICS AND COMPUTING↗

AMG Preconditioners based on parallel hybrid coarsening and multi-objective graph matching

We describe preliminary results from a multi-objective graph matching algorithm, in the coarsening step of an aggregation-based Algebraic MultiGrid (AMG) preconditioner, for solving large and sparse linear systems of equations on high-end parallel computers. We have two objectives. First, we wish to improve the convergence behavior of the AMG method when applied to highly anisotropic problems. Second, we wish to extend the parallel package \texttt{PSCToolkit} to exploit multi-threaded parallelism at the node level on multi-core processors. Our matching proposal balances the need to simultaneously compute high weights and large cardinalities by a new formulation of the weighted matching problem combining both these objectives using a parameter $\lambda$. We compute the matching by a parallel $2/3-\varepsilon$-approximation algorithm for maximum weight matchings. Results with the new matching algorithm show that for a suitable choice of the parameter $\lambda$ we compute effective preconditioners in the presence of anisotropy, i.e., smaller solve times, setup times, iterations counts, and operator complexity.

D'Ambra, Pasqua↗

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 of 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 massive 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 that will be explored in future work.

97 MATHEMATICS AND COMPUTING↗

Parameter identification for nonlinear aerodynamic systems

Parameter identification for nonlinear aerodynamic systems is examined. It is presumed that the underlying model can be arranged into an input/output (I/O) differential operator equation of a generic form. The algorithm estimation is especially efficient since the equation error can be integrated exactly given any I/O pair to obtain an algebraic function of the parameters. The algorithm for parameter identification was extended to the order determination problem for linear differential system. The degeneracy in a least squares estimate caused by feedback was addressed. A method of frequency analysis for determining the transfer function G(j omega) from transient I/O data was formulated using complex valued Fourier based modulating functions in contrast with the trigonometric modulating functions for the parameter estimation problem. A simulation result of applying the algorithm is given under noise-free conditions for a system with a low pass transfer function.

Pearson, Allan E.↗

Optimization and analysis of large chemical kinetic mechanisms using the solution mapping method - Combustion of methane

A method of systematic optimization, solution mapping, as applied to a large-scale dynamic model is presented. The basis of the technique is parameterization of model responses in terms of model parameters by simple algebraic expressions. These expressions are obtained by computer experiments arranged in a factorial design. The developed parameterized responses are then used in a joint multiparameter multidata-set optimization. A brief review of the mathematical background of the technique is given. The concept of active parameters is discussed. The technique is applied to determine an optimum set of parameters for a methane combustion mechanism. Five independent responses - comprising ignition delay times, pre-ignition methyl radical concentration profiles, and laminar premixed flame velocities - were optimized with respect to thirteen reaction rate parameters. The numerical predictions of the optimized model are compared to those computed with several recent literature mechanisms. The utility of the solution mapping technique in situations where the optimum is not unique is also demonstrated.

Frenklach, Michael↗