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NASA NTRS1997

Application of the Two-Step Filter to Process Ranging Measurements for Relative Navigation in an Elliptical Orbit

This estimator breaks a nonlinear estimation problem into a set of over determined 'first step' states which are linear in the observations and 'second step' states which are ultimately the states of interest. Linear estimation methods are applied to filter the observations and produce the optimal first step state estimate. The 'second step' states are obtained through iterative nonlinear parameter estimation considering the first step states as observations. It has been shown that this process exactly minimizes the least squares cost function for static problems and provides a better solution than the iterated extended Kalman filter (EKF) for dynamic problems. The two step filter is applied in this paper to process range and range rate measurements between the two spacecraft. Details of the application of the two step estimator to this problem will be given, highlighting the use of a test for ill-conditioned covariance estimates that can result from the first order covariance propagation. A comparison will be made between the performance of the two step filter and the IEKF.

Garrison, James L.↗

DOE OSTI2016

Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax{sup 2α} + Bx{sup 2}

The numerical solutions of the time independent Schrödinger equation of different one-dimensional potentials forms are sometime achieved by the asymptotic iteration method. Its importance appears, for example, on its efficiency to describe vibrational system in quantum mechanics. In this paper, the Airy function approach and the Numerov method have been used and presented to study the oscillator anharmonic potential V(x) = Ax{sup 2α} + Bx{sup 2}, (A>0, B<0), with (α = 2) for quadratic, (α =3) for sextic and (α =4) for octic anharmonic oscillators. The Airy function approach is based on the replacement of the real potential V(x) by a piecewise-linear potential v(x), while, the Numerov method is based on the discretization of the wave function on the x-axis. The first energies levels have been calculated and the wave functions for the sextic system have been evaluated. These specific values are unlimited by the magnitude of A, B and α. It’s found that the obtained results are in good agreement with the previous results obtained by the asymptotic iteration method for α =3.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2018

Solving linear equations with messenger-field and conjugate gradient techniques: An application to CMB data analysis

We discuss linear system solvers invoking a messenger-field and compare them with (preconditioned) conjugate gradient approaches. We show that the messenger-field techniques correspond to fixed point iterations of an appropriately preconditioned initial system of linear equations. We then argue that a conjugate gradient solver applied to the same preconditioned system, or equivalently a preconditioned conjugate gradient solver using the same preconditioner and applied to the original system, will in general ensure at least a comparable and typically better performance in terms of the number of iterations to convergence and time-to-solution. We illustrate our conclusions with two common examples drawn from the cosmic microwave background (CMB) data analysis: Wiener filtering and map-making. In addition, and contrary to the standard lore in the CMB field, we show that the performance of the preconditioned conjugate gradient solver can depend significantly on the starting vector. In conclusion, this observation seems of particular importance in the cases of map-making of high signal-to-noise ratio sky maps and therefore should be of relevance for the next generation of CMB experiments.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

The matrix product approximation for the dynamic cavity method

Stochastic dynamics of classical degrees of freedom, defined on vertices of locally tree-like graphs, can be studied in the framework of the dynamic cavity method which is exact for tree graphs. Such models correspond for example to spin-glass systems, Boolean networks, neural networks, and other technical, biological, and social networks. The central objects in the cavity method are edge messages—conditional probabilities of two vertex variable trajectories. In this paper, we discuss a rather pedagogical derivation for the dynamic cavity method, give a detailed account of the novel matrix product edge message (MPEM) algorithm for the solution of the dynamic cavity equation as introduced, and present optimizations and extensions. Matrix product approximations of the edge messages are constructed recursively in an iteration over time. Computation costs and precision can be tuned by controlling the matrix dimensions of the MPEM in truncations. Without truncations, the dynamics is exact. Data for Glauber–Ising dynamics shows a linear growth of computation costs in time. In contrast to Monte Carlo simulations, the approach has a much better error scaling. Hence, it gives for example access to low probability events and decaying observables like temporal correlations. Here, we discuss optimized truncation schemes and an extension that allows to capture models which have a continuum time limit.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1979

A viscous flow analysis of the tip vortex generation process

A viscous primary-secondary flow analysis designed for numerical solution by forward marching integration is applied to the tip vortex generation problem. Equations governing streamwise momentum, streamwise vorticity, and a gas law are solved in conjunction with equations governing scalar and vector surface potentials for the secondary velocities. The numerical method used combines a consistently split linearized block implicit (LBI) scheme for parabolic equations and a scalar iterative ADI scheme for elliptic equations. Computed results are presented for a wing of rectangular planform immersed in a high Reynolds number stream at 6 degrees incidence. The physical mechanism for generation of the tip vortex, as revealed by the analysis and computations, is discussed. Detailed flow results for the case of 6 degree incidence are included.

Shamroth, S. J.↗

DOE OSTI2015

A Bound- and Equality-Constrained Quadratic Programming Method for Image Reconstruction

Image deblurring is modeled as an ill-posed, linear inverse problem and can be written in terms of an optimization problem. By adding bound and equality constraints and/or penalty terms to the objective function, the problem is regularized and a viable solution exists; however, there are few methods available for solving such constrained problems. Here we formulate two regularized solutions into constrained optimization problems and adapt an iterative augmented Lagrangian solver. Results are shown on synthetic data in one- and two-dimensions, as well as on a radiograph used to calibrate the transmission curve of a pulsed power X-ray source at a U.S. Department of Energy radiography facility.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2016

Quantum cluster equilibrium model of N-methylformamide–water binary mixtures

The established quantum cluster equilibrium (QCE) approach is refined and applied to N-methylformamide (NMF) and its aqueous solution. The QCE method is split into two iterative cycles: one which converges to the liquid phase solution of the QCE equations and another which yields the gas phase. By comparing Gibbs energies, the thermodynamically stable phase at a given temperature and pressure is then chosen. The new methodology avoids metastable solutions and allows a different treatment of the mean-field interactions within the gas and liquid phases. These changes are of crucial importance for the treatment of binary mixtures. For the first time in a QCE study, the cis-trans-isomerism of a species (NMF) is explicitly considered. Cluster geometries and frequencies are calculated using density functional theory (DFT) and complementary coupled cluster single point energies are used to benchmark the DFT results. Independent of the selected quantum-chemical method, a large set of clusters is required for an accurate thermodynamic description of the binary mixture. The liquid phase of neat NMF is found to be dominated by the cyclic trans-NMF pentamer, which can be interpreted as a linear trimer that is stabilized by explicit solvation of two further NMF molecules. This cluster reflects the known hydrogen bond network preferences of neat NMF.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2014

Model based multivariable controller for large scale compression stations. Design and experimental validation on the LHC 18KW cryorefrigerator

In this paper, a multivariable model-based non-linear controller for Warm Compression Stations (WCS) is proposed. The strategy is to replace all the PID loops controlling the WCS with an optimally designed model-based multivariable loop. This new strategy leads to high stability and fast disturbance rejection such as those induced by a turbine or a compressor stop, a key-aspect in the case of large scale cryogenic refrigeration. The proposed control scheme can be used to have precise control of every pressure in normal operation or to stabilize and control the cryoplant under high variation of thermal loads (such as a pulsed heat load expected to take place in future fusion reactors such as those expected in the cryogenic cooling systems of the International Thermonuclear Experimental Reactor ITER or the Japan Torus-60 Super Advanced fusion experiment JT-60SA). The paper details how to set the WCS model up to synthesize the Linear Quadratic Optimal feedback gain and how to use it. After preliminary tuning at CEA-Grenoble on the 400W@1.8K helium test facility, the controller has been implemented on a Schneider PLC and fully tested first on the CERN's real-time simulator. Then, it was experimentally validated on a real CERN cryoplant. The efficiency of the solution is experimentally assessed using a reasonable operating scenario of start and stop of compressors and cryogenic turbines. This work is partially supported through the European Fusion Development Agreement (EFDA) Goal Oriented Training Program, task agreement WP10-GOT-GIRO.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

NASA NTRS1983

Iterative optimal subcritical aerodynamic design code including profile drag

A subcritical aerodynamic design computer code has been developed, which uses linearized aerodynamics along with sweep theory and airfoil data to obtain minimum total drag preliminary designs for multiple planform configurations. These optimum designs consist of incidence distributions yielding minimum total drag at design values of Mach number and lift and pitching moment coefficients. Linear lofting is used between airfoil stations. Solutions for isolated transport wings have shown that the solution is unique, and that including profile drag effects decreases tip loading and incidence relative to values obtained for minimum induced drag solutions. Further, including effects of variation of profile drag with Reynolds number can cause appreciable changes in the optimal design for tapered wings. Example solutions are also discussed for multiple planform configurations.

Kuhlman, J. M.↗

DOE OSTI2018

Expansion in higher harmonics of boson stars using a generalized Ruffini-Bonazzola approach. Part 1. Bound states

The method pioneered by Ruffini and Bonazzola (RB) to describe boson stars involves an expansion of the boson field which is linear in creation and annihilation operators. This expansion constitutes an exact solution to a non-interacting field theory, and has been used as a reasonable ansatz for an interacting one. In this work, we show how one can go beyond the RB ansatz towards an exact solution of the interacting operator Klein-Gordon equation, which can be solved iteratively to ever higher precision. Our Generalized Ruffini-Bonazzola approach takes into account contributions from nontrivial harmonic dependence of the wavefunction, using a sum of terms with energy $k\,E_0$, where $k\geq1$ and $E_0$ is the chemical potential of a single bound axion. The method critically depends on an expansion in a parameter $\Delta \equiv \sqrt{1-E_0{}^2/m^2}<1$, where $m$ is the mass of the boson. In the case of the axion potential, we calculate corrections which are relevant for axion stars in the transition or dense branches. We find with high precision the local minimum of the mass, $M_{min}\approx 463\,f^2/m$, at $\Delta\approx0.27$, where $f$ is the axion decay constant. This point marks the crossover from transition to dense branches of solutions, and a corresponding crossover from structural instability to stability.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2019

Approximate Green’s Function Coupled Cluster A Method Employing Effective Dimension Reduction

The Green’s function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structures of molecular structures and periodic systems, especially when electrons are strongly correlated. However, in order for the GFCC method to become an approach that can be routinely used in the electronic structure calculations, more robust linear algebraic techniques and approximations need to be employed to reduce its extremely high computational overhead. In our recent studies, we showed that the GFCC equation can be numerically solved by seeking solutions of multiple linear systems in the frequency regime of interest, which can be easily distributed in a massively parallel environment, and broaden the avenue for more state-of-the-art numerical linear algebraic tools associated with multiple linear systems to be kicked in to this field. In the present pursuit of this practice, we show a successful application of model order reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires a high resolution of spectral function, instead of solving GFCC linear equation in the original dimension for every single frequency point, an easily-solvable linear system model with reduced dimension can be built upon projecting the original GFCC linear system into a subspace, from which the interpolation and extrapolation can give a reasonable approximation to the results of the original GFCC linear equations. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some select frequencies inside the frequency regime. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The approach is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems including carbon monoxide, 1,3-butadiene, benzene, and adenine molecules. As a byproduct, the high quality linear system model is also found to be able to provide high quality initial guess for the existing linear solver to improve the convergence performance.

Peng, Bo↗

DOE OSTI2025

Accelerating eigenvalue computation for nuclear structure calculations via perturbative corrections

Subspace projection methods utilizing perturbative corrections have been proposed for computing the lowest few eigenvalues and corresponding eigenvectors of large Hamiltonian matrices. In this paper, we build upon these methods and introduce the term Subspace Projection with Perturbative Corrections (SPPC) method to refer to this approach. We tailor the SPPC for nuclear many-body Hamiltonians represented in a truncated configuration interaction subspace, i.e., the no-core shell model (NCSM). We use the hierarchical structure of the NCSM Hamiltonian to partition the Hamiltonian as the sum of two matrices. The first matrix corresponds to the Hamiltonian represented in a small configuration space, whereas the second is viewed as the perturbation to the first matrix. Eigenvalues and eigenvectors of the first matrix can be computed efficiently. Because of the split, perturbative corrections to the eigenvectors of the first matrix can be obtained efficiently from the solutions of a sequence of linear systems of equations defined in the small configuration space. These correction vectors can be combined with the approximate eigenvectors of the first matrix to construct a subspace from which more accurate approximations of the desired eigenpairs can be obtained. We show by numerical examples that the SPPC method can be more efficient than conventional iterative methods for solving large-scale eigenvalue problems such as the Lanczos, block Lanczos and the locally optimal block preconditioned conjugate gradient (LOBPCG) method. The method can also be combined with other methods to avoid convergence stagnation.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS2017

Accuracy/Computation Performance of a New Trilateration Scheme for GPS-Style Localization

We recently introduced a new geometric trilateration (GT) method for GPS-style positioning. Preliminary singlepoint analysis using simplistic error assumptions indicates that the new scheme delivers almost indistinguishable localization accuracy as the traditional Newton-Raphson (NR) approach. Also, the same computation procedure can be used to perform high-accuracy relative positioning between a reference vehicle and an arbitrary number of target vehicles. This scheme has the potential to enable a) new mission concepts in collaborative science, b) in-situ navigation services for human Mars missions, and c) lower cost and faster acquisition of GPS signals for consumer-grade GPS products. The new GT scheme differs from the NR scheme as follows: 1. The new scheme is derived from Pythagoras Theorem, whereas the NR method is based on the principle of linear regression. 2. The NR method uses the absolute locations (xi, yi, zi)’s of the GPS satellites as input to each step of the localization computation. The GT method uses the Directional Cosines Ui’s from Earth’s center to the GPS satellite Si. 3. Both the NR method and the GT method iterate to converge to a localized solution. In each iteration step, multiple matrix operations are performed. The NR method constructs a different matrix in each iterative step, thus requires performing a new set of matrix operations in each step. The GT scheme uses the same matrix in each iteration, thus requiring computing the matrix operations only once for all subsequent iterations. In this paper, we perform an in-depth comparison between the GT scheme and the NR method in terms of a) GPS localization accuracy in the GPS operation environment, b) its sensitivity with respect to systematic errors and random errors, and c) computation load required to converge to a localization solution.

Lee, Charles↗

DOE OSTI2015

Optimal Voltage Regulation for Unbalanced Distribution Networks Considering Distributed Energy Resources

With increasing penetration of distributed generation in the distribution networks (DN), the secure and optimal operation of DN has become an important concern. In this paper, an iterative quadratic constrained quadratic programming model to minimize voltage deviations and maximize distributed energy resource (DER) active power output in a three phase unbalanced distribution system is developed. The optimization model is based on the linearized sensitivity coefficients between controlled variables (e.g., node voltages) and control variables (e.g., real and reactive power injections of DERs). To avoid the oscillation of solution when it is close to the optimum, a golden search method is introduced to control the step size. Numerical simulations on modified IEEE 13 nodes test feeders show the efficiency of the proposed model. Compared to the results solved by heuristic search (harmony algorithm), the proposed model converges quickly to the global optimum.

distributed energy resources↗

NASA NTRS1986

An efficient method for solving the steady Euler equations

An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.

Liou, M. S.↗

NASA NTRS1996

Numerical Modeling of Spray Combustion with an Unstructured-Grid Method

The present unstructured-grid method follows strictly the basic finite volume forms of the conservation laws of the governing equations for the entire flow domain. High-order spatially accurate formulation has been employed for the numerical solutions of the Navier-Stokes equations. A two-equation k-epsilon turbulence model is also incorporated in the unstructured-grid solver. The convergence of the resulted linear algebraic equation is accelerated with preconditioned Conjugate Gradient method. A statistical spray combustion model has been incorporated into the present unstructured-grid solver. In this model, spray is represented by discrete particles, rather than by continuous distributions. A finite number of computational particles are used to predict a sample of total population of particles. Particle trajectories are integrated using their momentum and motion equations and particles exchange mass, momentum and energy with the gas within the computational cell in which they are located. The interaction calculations are performed simultaneously and eliminate global iteration for the two-phase momentum exchange. A transient spray flame in a high pressure combustion chamber is predicted and then the solution of liquid-fuel combusting flow with a rotating cup atomizer is presented and compared with the experimental data. The major conclusion of this investigation is that the unstructured-grid method can be employed to study very complicated flow fields of turbulent spray combustion. Grid adaptation can be easily achieved in any flow domain such as droplet evaporation and combustion zone. Future applications of the present model can be found in the full three-dimensional study of flow fields of gas turbine and liquid propulsion engine combustion chambers with multi-injectors.

Shang, H. M.↗

DOE OSTI2023

A fast Fourier transform-based solver for elastic micropolar composites

This work presents a spectral micromechanical formulation for obtaining the full-field and homogenized response of elastic micropolar composites. The algorithm relies on a coupled set of convolution integral equations for the micropolar strains, where periodic Green’s operators associated with a linear homogeneous reference medium are convolved with functions of the Cauchy and couple stress fields that encode the material’s heterogeneity, as well as any potential material nonlinearity. Such convolution integral equations take an algebraic form in the reciprocal Fourier space that can be solved iteratively. In this vein, the fast Fourier transform (FFT) algorithm is leveraged to accelerate the numerical solution, resulting in a mesh-free formulation in which the periodic unit cell representing the heterogeneous material can be discretized by a regular grid of pixels in two dimensions (or voxels in three dimensions). For verification, the numerical solutions obtained with the micropolar FFT solver are compared with analytical solutions for a matrix with a dilute circular inclusion subjected to plane strain loading. The developed computational framework is then used to study length-scale effects and effective (micropolar) moduli of composites with various topological configurations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Pressure stability in explicitly coupled simulations of poromechanics with application to CO 2 sequestration

We study in detail the pressure stabilizing effects of the non-iterated fixed-stress splitting in poromechanical problems which are nearly undrained and incompressible. When applied in conjunction with a spatial discretization which does not satisfy the discrete inf–sup condition, namely a mixed piecewise linear–piecewise constant spatial discretization, the explicit fixed-stress scheme can have a pressure stabilizing effect in transient problems. This effect disappears, however, upon time step refinement or the attainment of steady state. The interpretation of the scheme as an Augmented Lagrangian method similar to Uzawa iteration for incompressible flow helps explain these results. Moreover, due to the slowly evolving solution within undrained seal regions, we show that the explicit fixed-stress scheme requires very large time steps to reveal its pressure stabilizing effect in examples of geologic CO 2 sequestration. We note that large time steps can result in large errors in drained regions, such as the aquifer or reservoir regions of these examples, and can prevent convergence of nonlinear solvers in the case of multiphase flows, which can make the explicit scheme an unreliable source of pressure stabilization. We conclude by demonstrating that pressure jump stabilization is as effective in the explicit fixed-stress setting as in the fully implicit setting for undrained problems, while maintaining the stability and convergence of the fixed-stress split for drained problems.

58 GEOSCIENCES↗