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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.

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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
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 NTRS1984

Design of multivariable feedback control systems via spectral assignment using reduced-order models and reduced-order observers

The feasibility of using reduced order models and reduced order observers with eigenvalue/eigenvector assignment procedures is investigated. A review of spectral assignment synthesis procedures is presented. Then, a reduced order model which retains essential system characteristics is formulated. A constant state feedback matrix which assigns desired closed loop eigenvalues and approximates specified closed loop eigenvectors is calculated for the reduced order model. It is shown that the eigenvalue and eigenvector assignments made in the reduced order system are retained when the feedback matrix is implemented about the full order system. In addition, those modes and associated eigenvectors which are not included in the reduced order model remain unchanged in the closed loop full order system. The full state feedback design is then implemented by using a reduced order observer. It is shown that the eigenvalue and eigenvector assignments of the closed loop full order system rmain unchanged when a reduced order observer is used. The design procedure is illustrated by an actual design problem.

Mielke, R. R.↗
NASA NTRS1985

Design of multivariable feedback control systems via spectral assignment using reduced-order models and reduced-order observers

The feasibility of using reduced order models and reduced order observers with eigenvalue/eigenvector assignment procedures is investigated. A review of spectral assignment synthesis procedures is presented. Then, a reduced order model which retains essential system characteristics is formulated. A constant state feedback matrix which assigns desired closed loop eigenvalues and approximates specified closed loop eigenvectors is calculated for the reduced order model. It is shown that the eigenvalue and eigenvector assignments made in the reduced order system are retained when the feedback matrix is implemented about the full order system. In addition, those modes and associated eigenvectors which are not included in the reduced order model remain unchanged in the closed loop full order system. The fulll state feedback design is then implemented by using a reduced order observer. It is shown that the eigenvalue and eigenvector assignments of the closed loop full order system remain unchanged when a reduced order observer is used. The design procedure is illustrated by an actual design problem.

Mielke, R. R.↗
NASA NTRS1984

Description of sunspot cycles by orthogonal functions

Based on the principal component analysis technique and evidence for a 22-yr double-sunspot cycle periodicity. The time series of sunspot numbers is represented as a sum of mutually orthogonal eigenvectors in the time domain. It is shown that the first two eigenvectors account for about 90 percent of the cumulative 'signal power,' and that this is sufficient for reconstruction of the raw data curve. It is also noted that the second eigenvector behaves as the time derivative of the first, and that a phase-plane plot of these eigenvectors (i.e. a plot of a variable vs. its rate of change) suggests that the sun's sunspot cycle is driven by an oscillator; the implication is that, embedded within the sun, a chronometer is at work (e.g. Dicke, 1979).

Teuber, D. L.↗
NASA NTRS1987

Modal decoupling insensitivity

This note deals with the design of an output feedback control system such that selected closed-loop eigenvalues are insensitive, while the corresponding eigenvectors preserve a prescribed decoupling under parameter variation. This results in restricting specified elements of an eigenvector to be insensitive, rather than the entire eigenvector. This requirement is henceforth referred to as modal decoupling insensitivity. Sufficient conditions for modal decoupling insensitivity are derived. These conditions offer the advantage of increased flexibility in eigenvector assignment, but the constraint on the feedback matrix is more restrictive. A numerical example is presented to illustrate the design procedure.

Raman, K. V.↗
NASA NTRS1988

An O(N squared) method for computing the eigensystem of N by N symmetric tridiagonal matrices by the divide and conquer approach

An efficient method is proposed to solve the eigenproblem of N by N Symmetric Tridiagonal (ST) matrices. Unlike the standard eigensolvers which necessitate O(N cubed) operations to compute the eigenvectors of such ST matrices, the proposed method computes both the eigenvalues and eigenvectors with only O(N squared) operations. The method is based on serial implementation of the recently introduced Divide and Conquer (DC) algorithm. It exploits the fact that by O(N squared) of DC operations, one can compute the eigenvalues of N by N ST matrix and a finite number of pairs of successive rows of its eigenvector matrix. The rest of the eigenvectors--all of them or one at a time--are computed by linear three-term recurrence relations. Numerical examples are presented which demonstrate the superiority of the proposed method by saving an order of magnitude in execution time at the expense of sacrificing a few orders of accuracy.

Gill, Doron↗
NASA NTRS1990

Robust eigensystem assignment for second-order estimators

An approach for the robust eigensystem assignment of flexible structures using full state or output feedback is developed. Using the second-order dynamic equations, the approach can assign the eigenvalues of the system via velocity and displacement feedbacks, or acceleration and velocity feedbacks. The eigenvalues and eigenvectors of the system are assigned, via the second-order eigenvalue problem for the structural system, in two steps. First, an orthonormal basis spanning the attainable closed-loop eigenvector space corresponding to each desired closed-loop eigenvalue is generated using the Singular Value or QR decompositions. Second, a sequential procedure is used to choose a set of closed-loop eigenvectors that are as close as possible to the column space of a well-conditioned target matrix. Among the possible choices of the target matrix, the closest unitary matrix to the open-loop eigenvector matrix appears to be a suitable choice. A numerical example is given to illustrate the proposed algorithm.

Juang, Jer-Nan↗
NASA NTRS1990

Robust eigensystem assignment for second-order dynamic systems

A novel approach for the robust eigensystem assignment of flexible structures using full state or output feedback is developed. Using the second-order dynamic equations, the approach can assign the eigenvalues of the system via velocity and displacement feedbacks, or acceleration and velocity feedbacks. The eigenvalues and eigenvectors of the system are assigned, via the second-order eigenvalue problem for the structural system, in two steps. First, an orthonormal basis spanning the attainable closed-loop eigenvector space corresponding to each desired closed-loop eigenvalue is generated using the Singular Value or QR decompositions. Second, a sequential procedure is used to choose a set of closed-loop eigenvectors that are as close as possible to the column space of a well-conditioned target matrix. Among the possible choices of the target matrix, the closest unitary matrix to the open-loop eigenvector matrix appears to be a suitable choice. A numerical example is given to illustrate the proposed algorithm.

Juang, Jer-Nan↗
NASA NTRS1994

Gain weighted eigenspace assignment

This report presents the development of the gain weighted eigenspace assignment methodology. This provides a designer with a systematic methodology for trading off eigenvector placement versus gain magnitudes, while still maintaining desired closed-loop eigenvalue locations. This is accomplished by forming a cost function composed of a scalar measure of error between desired and achievable eigenvectors and a scalar measure of gain magnitude, determining analytical expressions for the gradients, and solving for the optimal solution by numerical iteration. For this development the scalar measure of gain magnitude is chosen to be a weighted sum of the squares of all the individual elements of the feedback gain matrix. An example is presented to demonstrate the method. In this example, solutions yielding achievable eigenvectors close to the desired eigenvectors are obtained with significant reductions in gain magnitude compared to a solution obtained using a previously developed eigenspace (eigenstructure) assignment method.

Davidson, John B.↗
NASA NTRS1997

The Topology of Symmetric Tensor Fields

Combinatorial topology, also known as "rubber sheet geometry", has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions.

Levin, Yingmei↗
NASA NTRS2014

Elastic Model Transitions: A Hybrid Approach Utilizing Quadratic Inequality Constrained Least Squares (LSQI) and Direct Shape Mapping (DSM)

A method for transitioning linear time invariant (LTI) models in time varying simulation is proposed that utilizes a hybrid approach for determining physical displacements by augmenting the original quadratically constrained least squares (LSQI) algorithm with Direct Shape Mapping (DSM) and modifying the energy constraints. The approach presented is applicable to simulation of the elastic behavior of launch vehicles and other structures that utilize discrete LTI finite element model (FEM) derived mode sets (eigenvalues and eigenvectors) that are propagated throughout time. The time invariant nature of the elastic data presents a problem of how to properly transition elastic states from the prior to the new model while preserving motion across the transition and ensuring there is no truncation or excitation of the system. A previous approach utilizes a LSQI algorithm with an energy constraint to effect smooth transitions between eigenvector sets with no requirement that the models be of similar dimension or have any correlation. This approach assumes energy is conserved across the transition, which results in significant non-physical transients due to changing quasi-steady state energy between mode sets, a phenomenon seen when utilizing a truncated mode set. The computational burden of simulating a full mode set is significant so a subset of modes is often selected to reduce run time. As a result of this truncation, energy between mode sets may not be constant and solutions across transitions could produce non-physical transients. In an effort to abate these transients an improved methodology was developed based on the aforementioned approach, but this new approach can handle significant changes in energy across mode set transitions. It is proposed that physical velocities due to elastic behavior be solved for using the LSQI algorithm, but solve for displacements using a two-step process that independently addresses the quasi-steady-state and non-steady-state contributions to the elastic displacement. For structures subject to large external forces, such as thrust or atmospheric drag, it is imperative to capture these forces when solving for elastic displacement. To simplify the mathematical formulation, assumptions are made regarding mass matrix normalization, constant external forcing, and constant viscous damping. These simplifications allow for direct solutions to the quasi-steady-state displacements through a process titled Direct Shape Mapping. DSM solves for the displacements using the eigenvalues of the elastic modes and the external forcing and returns a set of elastic displacements dictated by the eigenvectors of the post-transition mode set. For the non-steady-state contributions to displacement we formulate a LSQI problem that is constrained by energy of the non-steady state terms. The contributions from the quasi-steady-state and non-steady state solutions are then combined to obtain the physical displacements associated with the new set of eigenvectors. Results for the LSQI-DSM approach show significant reduction/complete removal of transients across mode set transitions while maintaining elastic motion from the prior state. For time propagation applications employing discrete elastic models that need to be transitioned in time and where running with full a full mode set is not feasible, the method developed offers a practical solution to simulating vehicle elasticity.

Hannan, Mike R.↗
DOE OSTI2024

Neutrino oscillations in matter using the adjugate of the Hamiltonian

We revisit neutrino oscillations in constant matter density for a number of different scenarios: three flavors with the standard Wolfenstein matter potential, four flavors with standard matter potential and three flavors with non-standard matter potentials. To calculate the oscillation probabilities for these scenarios one must determine the eigenvalues and eigenvectors of the Hamiltonians. We use a method for calculating the eigenvalues that is well known, determination of the zeros of determinant of matrix (λI - H), where H is the Hamiltonian, I the identity matrix and λ is a scalar. To calculate the associated eigenvectors we use a method that is little known in the particle physics community, the calculation of the adjugate (transpose of the cofactor matrix) of the same matrix, (λI - H). This method can be applied to any Hamiltonian, but provides a very simple way to determine the eigenvectors for neutrino oscillation in matter, independent of the complexity of the matter potential. This method can be trivially automated using the Faddeev–LeVerrier algorithm for numerical calculations. For the above scenarios we derive a number of quantities that are invariant of the matter potential, many are new such as the generalization of the Naumov–Harrison–Scott identity for four or more flavors of neutrinos. We also show how these matter potential independent quantities become matter potential dependent when off-diagonal non-standard matter effects are included.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1979

Principal component analysis of solar flares in the soft X-ray flux

The paper considers principal component analysis of solar flares in the soft X-ray flux, a technique for extracting the salient features from a mass of data. The method applies particularly to the analysis of nonstationary ensembles, and its computations require the evaluation of eigenvalues of matrices. The Eispack matrix eigen system routines were used to analyze full-disk proportional-counter data from the X-ray event analyzer which was part of the Skylab experiment. Empirical orthogonal functions were derived for events in the soft X-ray spectrum between 2.5 and 20 A during different time periods, indicating that about 90% of the cumulative power of each analyzed flare is contained in the largest eigenvector. The first two largest eigenvectors are sufficient for an empirical curve fit through the raw data and a characterization of solar flares in the soft X-ray flux, and power spectra of two largest eigenvectors reveal a reported periodicity of about 5 min.

Teuber, D. L.↗
NASA NTRS1983

Accelerating an iterative process by explicit annihilation

A slowly convergent stationary iterative process can be accelerated by explicitly annihilating (i.e., eliminating) the dominant eigenvector component of the error. The dominant eigenvalue or complex pair of eigenvalues can be estimated from the solution during the iteration. The corresponding eigenvector or complex pair of eigenvectors can then be annihilated by applying an explicit Richardson process over the basic iterative method. This can be done entirely in real arithmetic by analytically combining the complex conjugate annihilation steps. The technique is applied to an implicit algorithm for the calculation of two dimensional steady transonic flow over a circular cylinder using the equations of compressible inviscid gas dynamics. This demonstrates the use of explicit annihilation on a nonlinear problem.

Jespersen, D. C.↗
NASA NTRS1983

Optimal Landsat transforms for forest applications

Eleven transformations of data from four Landsat MSS channels were investigated to find if any of the transforms accentuated the separability of natural vegetation classes in regions of varying topographical relief. Attention was given to the divergence analysis and classification accuracy of information content of the eleven transforms and four channels. A useful scaling function was observed with the second eigenvector being the denominator in the divergence values obtained. The second eigenvector was found to reduce the effects of shadowing and differential illumination of vegetation signatures, thereby enhancing the divergence values. The highest accuracies in crop identification were provided by the averages of channels 4, 6, and 7 divided by the second eigenvector.

Logan, T. L.↗
NASA NTRS1985

Accelerating an iterative process by explicit annihilation

A slowly convergent stationary iterative process can be accelerated by explicitly annihilating (i.e., eliminating) the dominant eigenvector component of the error. The dominant eigenvalue or complex pair of eigenvalues can be estimated from the solution during the iteration. The corresponding eigenvector or complex pair of eigenvectors can then be annihilated by applying an explicit Richardson process over the basic iterative method. This can be done entirely in real arithmetic by analytically combining the complex conjugate annihilation steps. The technique is applied to an implicit algorithm for the calculation of two dimensional steady transonic flow over a circular cylinder using the equations of compressible inviscid gas dynamics. This demonstrates the use of explicit annihilation on a nonlinear problem.

Jespersen, D. C.↗
NASA NTRS1985

Interactive software for spectral assignment

A new interactive computer software package for eigenvalue/eigenvector assignment using constant state feedback is described. The package consists of ten subprograms, each associated with a specific design objective, accessible from a main control program. Using this package, primary design objectives of assigning eigenvalues and approximating eigenvectors are first achieved. Then secondary design objectives, including modification of specific eigenvector components, reduction in specified elements of the feedback gain matrix, and reduction in eigensystem sensitivity to changes in plant parameters, are addressed. These secondary objectives are achieved by a systematic modification of the assigned eigenvectors in a small region about the initial assignment. In addition, the program implements the use of spectral assignment procedures with reduced-order system models. Program modes are described and illustrated by numerical examples.

Mielke, R. R.↗
NASA NTRS1989

Flutter suppression using eigenspace freedoms to meet requirements

A constrained optimization methodology has been developed which allows specific use of eigensystem freedoms to meet design requirements. A subset of the available eigenvector freedoms was employed. The eigenvector freedoms associated with a particular closed-loop eigenvalue are coefficients of basis vectors which span the subspace in which that closed-loop vector must lie. Design requirements are included as a vector of inequality constraints. The procedure was successfully applied to develop an unscheduled controller which stabilizes symmetric flutter of an aeroelastic vehicle to a dynamic pressure 44 percent above the open-loop flutter point. The design process proceeded from full-state feedback to the inclusion of a full-order observer to the selection of an eighth-order controller which preserved the full-state sensitivity characteristics. Only a subset of the design freedoms was utilized (i.e., assuming full-state feedback only four out of 26 eigenvectors were used, and no variations were made in the closed-loop eigenvalues). Utilization of additional eigensystem freedoms could further improve the controller.

Adams, William M., Jr.↗