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At least 469 records · Page 26

An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs

The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L(sub 2) stability on a finite domain.

Warming, Robert F.↗

An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs

The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L (sub 2) stability on a finite domain.

Warming, Robert F.↗

Sturm-Liouville eigenproblems with an interior pole

The eigenvalues and eigenfunctions of self-adjoint Sturm-Liouville problems with a simple pole on the interior of an interval are investigated. Three general theorems are proved, and it is shown that as n approaches infinity, the eigenfunctions more and more closely resemble those of an ordinary Sturm-Liouville problem. The low-order modes differ significantly from those of a nonsingular eigenproblem in that both eigenvalues and eigenfunctions are complex, and the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. In addition, the WKB is shown to be accurate for all n, and all eigenvalues of a normal one-dimensional Sturm-Liouville equation with nonperiodic boundary conditions are well separated.

Boyd, J. P.↗

Impact of Radial Reflector Fidelity on Neutronics and Vessel Fluence Simulations

The Consortium for Advanced Simulation of Light Water Reactors is developing the Virtual Environment for Reactor Applications (VERA), and the MPACT code, which is the primary deterministic neutron transport solver in VERA, provides sub-pin level flux and power distributions as part of full-scale cycle depletion and analysis. In such calculations, an important aspect is the radial reflector treatment. To improve the fidelity of the radial reflector treatment, MPACT was extended to approximate the modeling of the reactor’s structural components such as the core shroud, barrel, neutron pads, and vessel. This work explores several modeling configurations with varying levels of fidelity and computational burden and assesses the importance of modeling fidelity on the eigenvalue and pin power distribution. Two two-dimensional (2-D) problems were analyzed to assess the impact on eigenvalue and pin power distributions with low-fidelity, coarse square cell reflector representations: (1) a Watts Bar Nuclear Plant Unit 1 (WBN1) quarter-core slice with depletion and (2) an AP1000 quarter-core slice. In this work, the analyses showed that the effect on eigenvalue is fairly small, but the effect on pin power is more pronounced, especially locally in the assemblies closest to the periphery, where the maximum pin power difference is nearly 3.5% in the AP1000 case. Two additional 2-D problems were used to assess the comparison between the low-fidelity coarse square cell treatment and a high-fidelity geometric representation that uses subpin material specification: (1) the same WBN1 quarter-core slice and (2) a representative model of the NuScale small modular reactor (SMR), which features a solid reflector design with moderator holes. These results demonstrate that even a coarse, low-fidelity representation adequately captures the necessary simulation characteristics. Last, these capabilities were applied to the 2-D WBN1 quarter-core depletion to assess the impact on vessel fluence using VeraShift. From adjoint calculations, pins along the periphery were observed to be of highest importance for fluence calculation, so the impact of the reflector representation in MPACT could theoretically substantially affect the predicted result. However, it was observed that the change in pin powers along the periphery minimally impacts the maximum vessel fluence with a difference within the statistical uncertainty but provides terrific insight on the sensitivity of the peripheral pins.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Relativistic levels of mesic atoms

We revisit the derivation of the covariant two-body scalar-fermion equation with a Coulomb interaction, presented in a previous paper. We show that it can be given the formal aspect of a Dirac equation, but for the fact that the eigenvalue is also contained in one of the coefficients and thus it is not linearly included. The discussion of the boundary value problem is therefore different, although some properties of the Dirac equation can be recovered in an approximation bringing back to the concept of reduced mass. We discuss a mixed analytic-numerical method of solution which allows to obtain very accurate results and we calculate the lowest levels, states and QED corrections for pionic and kaonic atoms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Sampling electronic structure quadratic unconstrained binary optimization problems (QUBOs) with Ocean and Mukai solvers

The most advanced D-Wave Advantage quantum annealer has 5000+ qubits, however, every qubit is connected to a small number of neighbors. As such, implementation of a fully-connected graph results in an order of magnitude reduction in qubit count. To compensate for the reduced number of qubits, one has to rely on special heuristic software such as qbsolv, the purpose of which is to decompose a large quadratic unconstrained binary optimization (QUBO) problem into smaller pieces that fit onto a quantum annealer. In this work, we compare the performance of the open-source qbsolv which is a part of the D-Wave Ocean tools and a new Mukai QUBO solver from Quantum Computing Inc. (QCI). The comparison is done for solving the electronic structure problem and is implemented in a classical mode (Tabu search techniques). The Quantum Annealer Eigensolver is used to map the electronic structure eigenvalue-eigenvector equation to a QUBO problem, solvable on a D-Wave annealer. We find that the Mukai QUBO solver outperforms the Ocean qbsolv with one to two orders of magnitude more accurate energies for all calculations done in the present work, both the ground and excited state calculations. This work stimulates the further development of software to assist in the utilization of modern quantum annealers.

97 MATHEMATICS AND COMPUTING↗

On eigenvectors of multiple eigenvalues obtained in NASTRAN

In the case of nonmultiple eigenvalues, each of the three real eigenvalue extraction methods available in NASTRAN will, for a given type of normalization, give essentially the same eigenvectors, but this is not so in the case of multiple eigenvalues. This discrepancy is explained and illustrated by considering the example of a NASTRAN demonstration problem that has both multiple and nonmultiple eigenvalues.

Pamidi, P. R.↗

Numerical pole assignment by eigenvalue Jacobian inversion

A numerical procedure for solving the linear pole placement problem is developed which operates by the inversion of an analytically determined eigenvalue Jacobian matrix. Attention is given to convergence characteristics and pathological situations. It is not concluded that the algorithm developed is suitable for computer-aided control system design with particular reference to the scan platform pointing control system for the Galileo spacecraft.

Sevaston, George E.↗

Stability and bifurcation of quasiparallel Alfven solitons

The inverse scattering transformation (IST) is used to study the one-parameter and two-parameter soliton families of the derivative nonlinear Schroedinger (DNLS) equation. The two-parameter soliton family is determined by the discrete complex eigenvalue spectrum of the Kaup-Newell scattering problem and the one-parameter soliton family corresponds to the discrete real eigenvalue spectrum. The structure of the IST is exploited to discuss the existence of discrete real eigenvalues and to prove their structural stability to perturbations of the initial conditions. Also, though the two-parameter soliton is structurally stable in general, it is shown that a perturbation of the initial conditions may change the two-parameter soliton into a degenerate soliton which, in turn, is structurally unstable. This degenerate, or double pole, soliton may bifurcate due to a perturbation of the initial conditions into a pair of one-parameter solitons. If the initial profile is on compact support, then this pair of one-parameter solitons must be compressive and rarefactive respectively. Finally, the Gelfand-Levitan equations appropriate for the double pole soliton are solved.

Hamilton, R. L.↗

Topology optimization of 3D photonic crystals with complete bandgaps

The design of photonic crystals with complete bandgaps has recently received considerable research focus for numerous reasons. This work leverages well-known nonlinear programming techniques to alleviate the non-smoothness caused by degenerate eigenvalues such that topology optimization problems can be solved with the open-source IPOPT software. A fully-vectorial plane wave expansion technique is used with an iterative eigensolver to efficiently predict dispersion properties of candidate structures. Nonlinear programming is employed to solve the inverse problem of designing three-dimensional periodic structures that exhibit complete two-dimensional (2D) and three-dimensional (3D) photonic bandgaps. Mesh refinement is performed to alleviate the large computational burden of designing and analyzing photonic crystals, and a periodic density filter is implemented to impose a minimum feature size for manufacturability considerations.

97 MATHEMATICS AND COMPUTING↗

On the flame spreading over a polymer surface

The characteristics of flame propagation along the interface between a solid fuel and a gaseous oxidizer are considered. It is found that difficulties in a study of this problem are related to the eigenvalue character of the relative velocity of the flame with respect to the oxidizer flow. In the proposed model of flame spreading the difficulties are overcome by retaining the ellipticity of the governing equations and by taking the surface pyrolysis into account. The solution procedure makes use of a sink-source approach and the Fourier-transform method. Flame propagation velocity data computed on the basis of the described approach are compared with experimental results.

Ohki, Y.↗

A summary of spectral synthesis procedures for multivariable systems

A new approach to the eigensystem assignment problem is presented. The approach utilizes a null-space formulation of the eigenvalue/eigenvector assignment problem to simultaneously realize arbitrary eigenvalue specifications, approximate desired modal behavior, and achieve low eigensystem sensitivity with respect to plant parameter variations. The methods are applied to the design of regulator and integral plus proportional servo control systems.

Liberty, S. R.↗

Efficient computation of the stability of three-dimensional compressible boundary layers

Methods for the computer analysis of the stability of three-dimensional compressible boundary layers are discussed and the user-oriented Compressible Stability Analysis (COSAL) computer code is described. The COSAL code uses a matrix finite-difference method for local eigenvalue solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used initial-value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed which may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black-box stability code. Sample stability calculations are presented for the boundary layer profiles of an LFC swept wing.

Malik, M. R.↗

Numerical Techniques for Scattering from Submerged Objects

To represent the final results in terms of matrices, one expands all appropriate physical quantities in terms of partial wave basis states. This includes expansions for the incident and scattered fields and the surface quantities. The method then utilizes the Huygen-Poincare integral representation for both the exterior and interior solutions, leading to the required matrix equations. One thus deals with matrix equations, the complexity of which depends on the nature of the problem. It is shown that in general a transition matrix T can be obtained relating the incident field A with the scattered field f having the form T = PQ(-1), where f = TA. The structure of Q can be quite complicated and can itself be composed of other matrix inversions such as arise from layered objects. Recent improvements in this method appropriate for a variety of physical problems are focused on, and on their implementation. Results are outlined from scattering simulations for very elongated submerged objects and resonance scattering from elastic solids and shells. The final improvement concerns eigenfunction expansions of surface terms, arising from solution of the interior problem, obtained via a preconditioning technique. This effectively reduces the problem to that of obtaining eigenvalues of a Hermitian operator. This formalism is reviewed for scattering from targets that are rigid, sound-soft, acoustic, elastic solids, elastic shells, and elastic layered objects. Two sets of the more interesting results are presented. The first concerns scattering from elongated objects, and the second to thin elastic spheroids.

Werby, M. F.↗

The finite element machine - An assessment of the impact of parallel computing on future finite element computations

The requirements of complex aerospace vehicles combined with the age of structural analysis systems enhance the need to advance technology toward a new generation of structural analysis capability. Recent and impeding advances in parallel and supercomputers provide the opportunity to significantly improve these structural analysis capabilities for large order finite element problems. Long-term research in parallel computing, associated with the NASA Finite Element Machine project, is discussed. The results show the potential of parallel computers to provide substantial increases in computation speed over sequential computers. Results are given for sample problems in the areas of eigenvalue analysis and transient response.

Fulton, R. E.↗

Linear stability analysis of three-dimensional compressible boundary layers

A compressible stability analysis computer code is developed. The code uses a matrix finite-difference method for local eigenvale solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used inital-value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed that may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black-box stability code. Sample stability calculations are presented for the boundary layer profiles of an LFC swept wing.

Malik, Mujeeb R.↗

Frequency domain synthesis of optimal inputs for adaptive identification and control

The input design problem of selecting appropriate inputs for use in SISO adaptive identification and model reference adaptive control algorithms is considered. Averaging theory is used to characterize the optimal inputs in the frequency domain. The design problem is formulated as an optimization problem which maximizes the smallest eigenvalue of the average information matrix over power constrained signals, and the global optimal solution is obtained using a convergent numerical algorithm. A bound on the frequency search range required in the design algorithm has been determined in terms of the desired performance.

Fu, Li-Chen↗

A code for linear stability analysis

A new spectral code, Linear.x, has been written in FORTRAN 77 for the analysis of the linear stability of some basic state. While being generic and unrelated to any particular physical problem, the code provides for various common tasks, including global (eigenvalue spectra), local (single eigenvalues and eigenfunctions), table (one-dimensional and multidimensional tables of eigenvalues), curve (curves in parameter space), and others. A specific problem can be defined as a set of files some of which are included at compile time, while definitions, tasks, and parameters are read during run time. The code is currently used for the stability analysis of compressible flows.

Herbert, Thorwald↗