Rounding error bounds by perturbation- condition analysis
Rounding error bounds by perturbation condition analysis, and conditioning for eigenvalue problem
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State space approach to mixed boundary value problems.
State space procedure for mixed boundary value problems applied to structural analysis, fluid mechanics and eigenvalues for deformable bodies
Chen, C. F.↗
Guyan Reduction Solutions Recycled for Improved Accuracy
Procedures for accurate condensing of eigenvalue solutions in structural analysis using Guyan reduction solutions
Levy, R.↗
Survey of methods for calculating sensitivity of general eigenproblems
A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.
Murthy, Durbha V.↗
Thermographic Imaging of the Space Shuttle During Re-Entry Using a Near Infrared Sensor
High resolution calibrated near infrared (NIR) imagery of the Space Shuttle Orbiter was obtained during hypervelocity atmospheric re-entry of the STS-119, STS-125, STS-128, STS-131, STS-132, STS-133, and STS-134 missions. This data has provided information on the distribution of surface temperature and the state of the airflow over the windward surface of the Orbiter during descent. The thermal imagery complemented data collected with onboard surface thermocouple instrumentation. The spatially resolved global thermal measurements made during the Orbiter s hypersonic re-entry will provide critical flight data for reducing the uncertainty associated with present day ground-to-flight extrapolation techniques and current state-of-the-art empirical boundary-layer transition or turbulent heating prediction methods. Laminar and turbulent flight data is critical for the validation of physics-based, semi-empirical boundary-layer transition prediction methods as well as stimulating the validation of laminar numerical chemistry models and the development of turbulence models supporting NASA s next-generation spacecraft. In this paper we provide details of the NIR imaging system used on both air and land-based imaging assets. The paper will discuss calibrations performed on the NIR imaging systems that permitted conversion of captured radiant intensity (counts) to temperature values. Image processing techniques are presented to analyze the NIR data for vignetting distortion, best resolution, and image sharpness. Keywords: HYTHIRM, Space Shuttle thermography, hypersonic imaging, near infrared imaging, histogram analysis, singular value decomposition, eigenvalue image sharpness
Zalameda, Joseph N.↗
Stability analysis of nonlinear two-grid method for multigroup neutron diffusion problems
We present theoretical analysis of a nonlinear acceleration method for solving multigroup neutron diffusion problems. This method is formulated with two energy grids that are defined by (i) fine-energy groups structure and (ii) coarse grid with just a single energy group. The coarse-grid equations are derived by averaging of the multigroup diffusion equations over energy. The method uses a nonlinear prolongation operator. We perform stability analysis of iteration algorithms for inhomogeneous (fixed-source) and eigenvalue neutron diffusion problems. To apply Fourier analysis the equations of the method are linearized about solutions of infinite-medium problems. The developed analysis enables us to predict convergence properties of this two-grid method in different types of problems. Numerical results of problems in 2D Cartesian geometry are presented to confirm theoretical predictions.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
State space parameterization of explosive eigenvalues during autoignition
Explosive modes such as ignition and extinction are characterized by an eigenvalue of the chemical Jacobian matrix with positive real part, representing the transient instability of chain-branching chemistry and thermal feedback. Formation and eigen-decomposition of the Jacobian matrix are expensive operations whose cost increases cubically with chemical mechanism size. As an alternative to directly computing the eigenvalues of the Jacobian, we explore principal component analysis (PCA) along with nonlinear regression as a methodology to parameterize the eigenvalues by state variables (or linear combinations thereof). We evaluate this modeling strategy using homogeneous autoignition data on two different applications: pseudotransient continuation (Ψtc)-based ODE solvers and chemical explosive mode analysis (CEMA). Results indicate that the PCA-based parameterization of the eigenvalues appears feasible for Ψtc solvers in autoignition calculations over a range of temperatures and pressures. Our results also show that eigenvalue models are capable of tracking sharp discontinuities (such as ignition or extinction) in the eigenvalue for computational flame diagnostics such as CEMA.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Sensitivity analysis of hydrodynamic stability operators
The eigenvalue sensitivity for hydrodynamic stability operators is investigated. Classical matrix perturbation techniques as well as the concept of epsilon-pseudoeigenvalues are applied to show that parts of the spectrum are highly sensitive to small perturbations. Applications are drawn from incompressible plane Couette, trailing line vortex flow and compressible Blasius boundary layer flow. Parametric studies indicate a monotonically increasing effect of the Reynolds number on the sensitivity. The phenomenon of eigenvalue sensitivity is due to the non-normality of the operators and their discrete matrix analogs and may be associated with large transient growth of the corresponding initial value problem.
Schmid, Peter J.↗
Integrated tools for control-system analysis
The basic functions embedded within a user friendly software package (MATRIXx) are used to provide a high level systems approach to the analysis of linear control systems. Various control system analysis configurations are assembled automatically to minimize the amount of work by the user. Interactive decision making is incorporated via menu options and at selected points, such as in the plotting section, by inputting data. There are five evaluations such as the singular value robustness test, singular value loop transfer frequency response, Bode frequency response, steady-state covariance analysis, and closed-loop eigenvalues. Another section describes time response simulations. A time response for random white noise disturbance is available. The configurations and key equations used for each type of analysis, the restrictions that apply, the type of data required, and an example problem are described. One approach for integrating the design and analysis tools is also presented.
Ostroff, Aaron J.↗
Vibration and local instability of thermally stressed plates
The vibration and buckling of a double wedge square cantilever plate has been investigated. It is shown that the free vibration modes, which occur at Delta T ref = 0, transition into the buckled modes which occur at Delta T ref = Delta T ref-cr for the respective mode. Delta T ref-cr for a particular mode is defined as the magnitude of thermal load at which the frequency of the particular mode vanishes. The analysis yields the same number of buckling eigenvalues and buckling modes as there are vibration eigenvalues and vibration modes. Gradual application of the load in the analysis permits the change in each vibration frequency of interest and its associated mode to be followed up to the load at which the frequency of the mode becomes zero. This constitutes the limit of linear theory. As the load is increased, the thin edges of the plate begin to deform during vibration. This local deformation, which begins in the vibration mode, is shown to transition into the phenomena of local edge buckling at Delta T ref-cr for the mode.
Bailey, C. D.↗
Multiresolution and Explicit Methods for Vector Field Analysis and Visualization
We first report on our current progress in the area of explicit methods for tangent curve computation. The basic idea of this method is to decompose the domain into a collection of triangles (or tetrahedra) and assume linear variation of the vector field over each cell. With this assumption, the equations which define a tangent curve become a system of linear, constant coefficient ODE's which can be solved explicitly. There are five different representation of the solution depending on the eigenvalues of the Jacobian. The analysis of these five cases is somewhat similar to the phase plane analysis often associate with critical point classification within the context of topological methods, but it is not exactly the same. There are some critical differences. Moving from one cell to the next as a tangent curve is tracked, requires the computation of the exit point which is an intersection of the solution of the constant coefficient ODE and the edge of a triangle. There are two possible approaches to this root computation problem. We can express the tangent curve into parametric form and substitute into an implicit form for the edge or we can express the edge in parametric form and substitute in an implicit form of the tangent curve. Normally the solution of a system of ODE's is given in parametric form and so the first approach is the most accessible and straightforward. The second approach requires the 'implicitization' of these parametric curves. The implicitization of parametric curves can often be rather difficult, but in this case we have been successful and have been able to develop algorithms and subsequent computer programs for both approaches. We will give these details along with some comparisons in a forthcoming research paper on this topic.
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Design, analysis and numerical experiments for the virtual element p and hp approximations of elliptic eigenvalue problems
We discuss the p- and the hp-versions of the virtual element method for the approximation of the eigenvalues of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schr¨odinger equation with a pseudo-potential term. We present in details the analysis of the p-version of the method, proving the exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the hp-spaces.
97 MATHEMATICS AND COMPUTING↗
Wave-turbulence dynamics in the stably stratified boundary layer
New data obtained at the Boulder Atmospheric Observatory are analyzed to obtain separation of wave, turbulence, and mean field necessary for a complete treatment of wave-turbulence interaction. The data were compared with a linear stability analysis of the background atmospheric state, showing good agreement between measured wave parameters (such as wavelength, period, and vector phase velocity) and the eigenvalues of the linear solution. The analysis of the budgets of wave heat flux and temperature variance revealed the essential role of wave-turbulence interaction in maintaining a large amplitude temperature wave and countergradient heat flux. A mechanism for the maintenance of turbulence by waves in strongly stratified boundary layers is described, which emphasizes that the time-mean Richardson number is an irrelevant parameter at such times.
Einaudi, F.↗
Eigenvalues of Rectangular Waveguide Using FEM With Hybrid Elements
A finite element analysis using hybrid triangular-rectangular elements is developed to estimate eigenvalues of a rectangular waveguide. Use of rectangular vector-edge finite elements in the vicinity of the PEC boundary and triangular elements in the interior region more accurately models the physical nature of the electromagnetic field, and consequently quicken the convergence.
Deshpande, Manohar D.↗
Analysis of structural correlations in a model binary 3D liquid through the eigenvalues and eigenvectors of the atomic stress tensors
It is possible to associate with every atom or molecule in a liquid its own atomic stress tensor. These atomic stress tensors can be used to describe liquids’ structures and to investigate the connection between structural and dynamic properties. In particular, atomic stresses allow to address atomic scale correlations relevant to the Green-Kubo expression for viscosity. Previously correlations between the atomic stresses of different atoms were studied using the Cartesian representation of the stress tensors or the representation based on spherical harmonics. In this paper we address structural correlations in a 3D model binary liquid using the eigenvalues and eigenvectors of the atomic stress tensors. This approach allows to interpret correlations relevant to the Green-Kubo expression for viscosity in a simple geometric way. On decrease of temperature the changes in the relevant stress correlation function between different atoms are significantly more pronounced than the changes in the pair density function. We demonstrate that this behaviour originates from the orientational correlations between the eigenvectors of the atomic stress tensors. We also found correlations between the eigenvalues of the same atomic stress tensor. For the studied system, with purely repulsive interactions between the particles, the eigenvalues of every atomic stress tensor are positive and they can be ordered: λ{sub 1} ≥ λ{sub 2} ≥ λ{sub 3} ≥ 0. We found that, for the particles of a given type, the probability distributions of the ratios (λ{sub 2}/λ{sub 1}) and (λ{sub 3}/λ{sub 2}) are essentially identical to each other in the liquids state. We also found that λ{sub 2} tends to be equal to the geometric average of λ{sub 1} and λ{sub 3}. In our view, correlations between the eigenvalues may represent “the Poisson ratio effect” at the atomic scale.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential
The paper is concerned with the spectral properties of second- order differential operators defined by periodic and quasi-periodic boundary conditions. We obtain asymptotic formulae for the eigenvalues, derive estimates of projections, give estimates for the equiconvergence of spectral decompositions, find sufficient conditions for operators to be spectral, and write down an asymptotic expansion for the semigroup of operators generated by the negative of the differential operator under consideration. Our estimates involve coefficients of the Fourier potential. The main results of the paper are obtained by using the method of similar operators. Bibliography: 34 titles. (paper)
97 MATHEMATICS AND COMPUTING↗
A description of the computer program STIFEIG for structural dynamic analysis
FORTRAN computer program for stiffness matrix eigenvalue calculation
Hou, S. N.↗