Solution of an eigenvalue problem for the Laplace operator on a spherical surface. M.S. Thesis - Maryland Univ.

NASA Technical Reports Server (NTRS)

Walden, H.

1974-01-01

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (also referred to as the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two specific types of spherical surface domains are considered: (1) the interior of a spherical triangle, i.e., the region bounded by arcs of three great circles, and (2) the exterior of a great circle arc extending for less than pi radians on the sphere (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is derived. The symmetric (generally five-point) finite difference equations that develop are written in matrix form and then solved by the iterative method of point successive overrelaxation. Upon convergence of this iterative method, the fundamental eigenvalue is approximated by iteration utilizing the power method as applied to the finite Rayleigh quotient.

Sturm-Liouville eigenproblems with an interior pole

NASA Technical Reports Server (NTRS)

Boyd, J. P.

1981-01-01

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.

Fourth-order convergence of a compact scheme for the one-dimensional biharmonic equation

NASA Astrophysics Data System (ADS)

Fishelov, D.; Ben-Artzi, M.; Croisille, J.-P.

2012-09-01

The convergence of a fourth-order compact scheme to the one-dimensional biharmonic problem is established in the case of general Dirichlet boundary conditions. The compact scheme invokes value of the unknown function as well as Pade approximations of its first-order derivative. Using the Pade approximation allows us to approximate the first-order derivative within fourth-order accuracy. However, although the truncation error of the discrete biharmonic scheme is of fourth-order at interior point, the truncation error drops to first-order at near-boundary points. Nonetheless, we prove that the scheme retains its fourth-order (optimal) accuracy. This is done by a careful inspection of the matrix elements of the discrete biharmonic operator. A number of numerical examples corroborate this effect. We also present a study of the eigenvalue problem uxxxx = νu. We compute and display the eigenvalues and the eigenfunctions related to the continuous and the discrete problems. By the positivity of the eigenvalues, one can deduce the stability of of the related time-dependent problem ut = -uxxxx. In addition, we study the eigenvalue problem uxxxx = νuxx. This is related to the stability of the linear time-dependent equation uxxt = νuxxxx. Its continuous and discrete eigenvalues and eigenfunction (or eigenvectors) are computed and displayed graphically.

Initial values for the integration scheme to compute the eigenvalues for propagation in ducts

NASA Technical Reports Server (NTRS)

Eversman, W.

1977-01-01

A scheme for the calculation of eigenvalues in the problem of acoustic propagation in a two-dimensional duct is described. The computation method involves changing the coupled transcendental nonlinear algebraic equations into an initial value problem involving a nonlinear ordinary differential equation. The simplest approach is to use as initial values the hardwall eigenvalues and to integrate away from these values as the admittance varies from zero to its actual value with a linear variation. The approach leads to a powerful root finding routine capable of computing the transverse and axial wave numbers for two-dimensional ducts for any frequency, lining, admittance and Mach number without requiring initial guesses or starting points.

Covariance expressions for eigenvalue and eigenvector problems

NASA Astrophysics Data System (ADS)

Liounis, Andrew J.

There are a number of important scientific and engineering problems whose solutions take the form of an eigenvalue--eigenvector problem. Some notable examples include solutions to linear systems of ordinary differential equations, controllability of linear systems, finite element analysis, chemical kinetics, fitting ellipses to noisy data, and optimal estimation of attitude from unit vectors. In many of these problems, having knowledge of the eigenvalue and eigenvector Jacobians is either necessary or is nearly as important as having the solution itself. For instance, Jacobians are necessary to find the uncertainty in a computed eigenvalue or eigenvector estimate. This uncertainty, which is usually represented as a covariance matrix, has been well studied for problems similar to the eigenvalue and eigenvector problem, such as singular value decomposition. There has been substantially less research on the covariance of an optimal estimate originating from an eigenvalue-eigenvector problem. In this thesis we develop two general expressions for the Jacobians of eigenvalues and eigenvectors with respect to the elements of their parent matrix. The expressions developed make use of only the parent matrix and the eigenvalue and eigenvector pair under consideration. In addition, they are applicable to any general matrix (including complex valued matrices, eigenvalues, and eigenvectors) as long as the eigenvalues are simple. Alongside this, we develop expressions that determine the uncertainty in a vector estimate obtained from an eigenvalue-eigenvector problem given the uncertainty of the terms of the matrix. The Jacobian expressions developed are numerically validated with forward finite, differencing and the covariance expressions are validated using Monte Carlo analysis. Finally, the results from this work are used to determine covariance expressions for a variety of estimation problem examples and are also applied to the design of a dynamical system.

Multigrid method for stability problems

NASA Technical Reports Server (NTRS)

Taasan, Shlomo

1988-01-01

The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are being solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a non-standard way in which the right hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relization on the finest level.

Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree-Fock.

PubMed

Hesselmann, Andreas; Görling, Andreas

2011-01-21

A recently introduced time-dependent exact-exchange (TDEXX) method, i.e., a response method based on time-dependent density-functional theory that treats the frequency-dependent exchange kernel exactly, is reformulated. In the reformulated version of the TDEXX method electronic excitation energies can be calculated by solving a linear generalized eigenvalue problem while in the original version of the TDEXX method a laborious frequency iteration is required in the calculation of each excitation energy. The lowest eigenvalues of the new TDEXX eigenvalue equation corresponding to the lowest excitation energies can be efficiently obtained by, e.g., a version of the Davidson algorithm appropriate for generalized eigenvalue problems. Alternatively, with the help of a series expansion of the new TDEXX eigenvalue equation, standard eigensolvers for large regular eigenvalue problems, e.g., the standard Davidson algorithm, can be used to efficiently calculate the lowest excitation energies. With the help of the series expansion as well, the relation between the TDEXX method and time-dependent Hartree-Fock is analyzed. Several ways to take into account correlation in addition to the exact treatment of exchange in the TDEXX method are discussed, e.g., a scaling of the Kohn-Sham eigenvalues, the inclusion of (semi)local approximate correlation potentials, or hybrids of the exact-exchange kernel with kernels within the adiabatic local density approximation. The lowest lying excitations of the molecules ethylene, acetaldehyde, and pyridine are considered as examples.

A few shape optimization results for a biharmonic Steklov problem

NASA Astrophysics Data System (ADS)

Buoso, Davide; Provenzano, Luigi

2015-09-01

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue.

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

NASA Astrophysics Data System (ADS)

Shirvany, Yazdan; Hayati, Mohsen; Moradian, Rostam

2008-12-01

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.

An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs

NASA Technical Reports Server (NTRS)

Warming, Robert F.; Beam, Richard M.

1989-01-01

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.

EvArnoldi: A New Algorithm for Large-Scale Eigenvalue Problems.

PubMed

Tal-Ezer, Hillel

2016-05-19

Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrödinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK ( Sorensen , D. C. Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations ; Springer , 1997 ; Lehoucq , R. B. ; Sorensen , D. C. SIAM Journal on Matrix Analysis and Applications 1996 , 17 , 789 ; Calvetti , D. ; Reichel , L. ; Sorensen , D. C. Electronic Transactions on Numerical Analysis 1994 , 2 , 21 ) (or Eigs of Matlab) but significantly simpler.

A comparison of matrix methods for calculating eigenvalues in acoustically lined ducts

NASA Technical Reports Server (NTRS)

Watson, W.; Lansing, D. L.

1976-01-01

Three approximate methods - finite differences, weighted residuals, and finite elements - were used to solve the eigenvalue problem which arises in finding the acoustic modes and propagation constants in an absorptively lined two-dimensional duct without airflow. The matrix equations derived for each of these methods were solved for the eigenvalues corresponding to various values of wall impedance. Two matrix orders, 20 x 20 and 40 x 40, were used. The cases considered included values of wall admittance for which exact eigenvalues were known and for which several nearly equal roots were present. Ten of the lower order eigenvalues obtained from the three approximate methods were compared with solutions calculated from the exact characteristic equation in order to make an assessment of the relative accuracy and reliability of the three methods. The best results were given by the finite element method using a cubic polynomial. Excellent accuracy was consistently obtained, even for nearly equal eigenvalues, by using a 20 x 20 order matrix.

The Schrodinger Eigenvalue March

ERIC Educational Resources Information Center

Tannous, C.; Langlois, J.

2011-01-01

A simple numerical method for the determination of Schrodinger equation eigenvalues is introduced. It is based on a marching process that starts from an arbitrary point, proceeds in two opposite directions simultaneously and stops after a tolerance criterion is met. The method is applied to solving several 1D potential problems including symmetric…

A systematic linear space approach to solving partially described inverse eigenvalue problems

NASA Astrophysics Data System (ADS)

Hu, Sau-Lon James; Li, Haujun

2008-06-01

Most applications of the inverse eigenvalue problem (IEP), which concerns the reconstruction of a matrix from prescribed spectral data, are associated with special classes of structured matrices. Solving the IEP requires one to satisfy both the spectral constraint and the structural constraint. If the spectral constraint consists of only one or few prescribed eigenpairs, this kind of inverse problem has been referred to as the partially described inverse eigenvalue problem (PDIEP). This paper develops an efficient, general and systematic approach to solve the PDIEP. Basically, the approach, applicable to various structured matrices, converts the PDIEP into an ordinary inverse problem that is formulated as a set of simultaneous linear equations. While solving simultaneous linear equations for model parameters, the singular value decomposition method is applied. Because of the conversion to an ordinary inverse problem, other constraints associated with the model parameters can be easily incorporated into the solution procedure. The detailed derivation and numerical examples to implement the newly developed approach to symmetric Toeplitz and quadratic pencil (including mass, damping and stiffness matrices of a linear dynamic system) PDIEPs are presented. Excellent numerical results for both kinds of problem are achieved under the situations that have either unique or infinitely many solutions.

Numerical solution of quadratic matrix equations for free vibration analysis of structures

NASA Technical Reports Server (NTRS)

Gupta, K. K.

1975-01-01

This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.

A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

NASA Astrophysics Data System (ADS)

Willert, Jeffrey; Park, H.; Knoll, D. A.

2014-10-01

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton-Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.

Root finding in the complex plane for seismo-acoustic propagation scenarios with Green's function solutions.

PubMed

McCollom, Brittany A; Collis, Jon M

2014-09-01

A normal mode solution to the ocean acoustic problem of the Pekeris waveguide with an elastic bottom using a Green's function formulation for a compressional wave point source is considered. Analytic solutions to these types of waveguide propagation problems are strongly dependent on the eigenvalues of the problem; these eigenvalues represent horizontal wavenumbers, corresponding to propagating modes of energy. The eigenvalues arise as singularities in the inverse Hankel transform integral and are specified by roots to a characteristic equation. These roots manifest themselves as poles in the inverse transform integral and can be both subtle and difficult to determine. Following methods previously developed [S. Ivansson et al., J. Sound Vib. 161 (1993)], a root finding routine has been implemented using the argument principle. Using the roots to the characteristic equation in the Green's function formulation, full-field solutions are calculated for scenarios where an acoustic source lies in either the water column or elastic half space. Solutions are benchmarked against laboratory data and existing numerical solutions.

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

NASA Astrophysics Data System (ADS)

Plestenjak, Bor; Gheorghiu, Călin I.; Hochstenbach, Michiel E.

2015-10-01

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.

A nonperturbative light-front coupled-cluster method

NASA Astrophysics Data System (ADS)

Hiller, J. R.

2012-10-01

The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential-operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from standard coupled-cluster theory, to obtain form factors and other observables.

Physics, stability, and dynamics of supply networks

NASA Astrophysics Data System (ADS)

Helbing, Dirk; Lämmer, Stefan; Seidel, Thomas; Šeba, Pétr; Płatkowski, Tadeusz

2004-12-01

We show how to treat supply networks as physical transport problems governed by balance equations and equations for the adaptation of production speeds. Although the nonlinear behavior is different, the linearized set of coupled differential equations is formally related to those of mechanical or electrical oscillator networks. Supply networks possess interesting features due to their complex topology and directed links. We derive analytical conditions for absolute and convective instabilities. The empirically observed “bullwhip effect” in supply chains is explained as a form of convective instability based on resonance effects. Moreover, it is generalized to arbitrary supply networks. Their related eigenvalues are usually complex, depending on the network structure (even without loops). Therefore, their generic behavior is characterized by damped or growing oscillations. We also show that regular distribution networks possess two negative eigenvalues only, but perturbations generate a spectrum of complex eigenvalues.

Solving Differential Equations Using Modified Picard Iteration

ERIC Educational Resources Information Center

Robin, W. A.

2010-01-01

Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

Calculation of normal modes of the closed waveguides in general vector case

NASA Astrophysics Data System (ADS)

Malykh, M. D.; Sevastianov, L. A.; Tiutiunnik, A. A.

2018-04-01

The article is devoted to the calculation of normal modes of the closed waveguides with an arbitrary filling ɛ, μ in the system of computer algebra Sage. Maxwell equations in the cylinder are reduced to the system of two bounded Helmholtz equations, the notion of weak solution of this system is given and then this system is investigated as a system of ordinary differential equations. The normal modes of this system are an eigenvectors of a matrix pencil. We suggest to calculate the matrix elements approximately and to truncate the matrix by usual way but further to solve the truncated eigenvalue problem exactly in the field of algebraic numbers. This approach allows to keep the symmetry of the initial problem and in particular the multiplicity of the eigenvalues. In the work would be presented some results of calculations.

Comments on numerical solution of boundary value problems of the Laplace equation and calculation of eigenvalues by the grid method

NASA Technical Reports Server (NTRS)

Lyusternik, L. A.

1980-01-01

The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners.

On the cross-stream spectral method for the Orr-Sommerfeld equation

NASA Technical Reports Server (NTRS)

Zorumski, William E.; Hodge, Steven L.

1993-01-01

Cross-stream models are defined as solutions to the Orr-Sommerfeld equation which are propagating normal to the flow direction. These models are utilized as a basis for a Hilbert space to approximate the spectrum of the Orr-Sommerfeld equation with plane Poiseuille flow. The cross-stream basis leads to a standard eigenvalue problem for the frequencies of Poiseuille flow instability waves. The coefficient matrix in the eigenvalue problem is shown to be the sum of a real matrix and a negative-imaginary diagonal matrix which represents the frequencies of the cross-stream modes. The real coefficient matrix is shown to approach a Toeplitz matrix when the row and column indices are large. The Toeplitz matrix is diagonally dominant, and the diagonal elements vary inversely in magnitude with diagonal position. The Poiseuille flow eigenvalues are shown to lie within Gersgorin disks with radii bounded by the product of the average flow speed and the axial wavenumber. It is shown that the eigenvalues approach the Gersgorin disk centers when the mode index is large, so that the method may be used to compute spectra with an essentially unlimited number of elements. When the mode index is large, the real part of the eigenvalue is the product of the axial wavenumber and the average flow speed, and the imaginary part of the eigen value is identical to the corresponding cross-stream mode frequency. The cross-stream method is numerically well-conditioned in comparison to Chebyshev based methods, providing equivalent accuracy for small mode indices and superior accuracy for large indices.

The method of Ritz applied to the equation of Hamilton. [for pendulum systems

NASA Technical Reports Server (NTRS)

Bailey, C. D.

1976-01-01

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum

NASA Astrophysics Data System (ADS)

Guarnieri, F.; Moon, W.; Wettlaufer, J. S.

2017-09-01

Motivated by a problem in climate dynamics, we investigate the solution of a Bessel-like process with a negative constant drift, described by a Fokker-Planck equation with a potential V (x ) =-[b ln(x ) +a x ] , for b >0 and a <0 . The problem belongs to a family of Fokker-Planck equations with logarithmic potentials closely related to the Bessel process that has been extensively studied for its applications in physics, biology, and finance. The Bessel-like process we consider can be solved by seeking solutions through an expansion into a complete set of eigenfunctions. The associated imaginary-time Schrödinger equation exhibits a mix of discrete and continuous eigenvalue spectra, corresponding to the quantum Coulomb potential describing the bound states of the hydrogen atom. We present a technique to evaluate the normalization factor of the continuous spectrum of eigenfunctions that relies solely upon their asymptotic behavior. We demonstrate the technique by solving the Brownian motion problem and the Bessel process both with a constant negative drift. We conclude with a comparison to other analytical methods and with numerical solutions.

Siewert solutions of transcendental equations, generalized Lambert functions and physical applications

NASA Astrophysics Data System (ADS)

Barsan, Victor

2018-05-01

Several classes of transcendental equations, mainly eigenvalue equations associated to non-relativistic quantum mechanical problems, are analyzed. Siewert's systematic approach of such equations is discussed from the perspective of the new results recently obtained in the theory of generalized Lambert functions and of algebraic approximations of various special or elementary functions. Combining exact and approximate analytical methods, quite precise analytical outputs are obtained for apparently untractable problems. The results can be applied in quantum and classical mechanics, magnetism, elasticity, solar energy conversion, etc.

Accelerating molecular property calculations with nonorthonormal Krylov space methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Furche, Filipp; Krull, Brandon T.; Nguyen, Brian D.

Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remainmore » small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computations of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.« less

Accelerating molecular property calculations with nonorthonormal Krylov space methods

DOE PAGES

Furche, Filipp; Krull, Brandon T.; Nguyen, Brian D.; ...

2016-05-03

Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remainmore » small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computations of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.« less

Extension of Nikiforov-Uvarov method for the solution of Heun equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Karayer, H., E-mail: hale.karayer@gmail.com; Demirhan, D.; Büyükkılıç, F.

2015-06-15

We report an alternative method to solve second order differential equations which have at most four singular points. This method is developed by changing the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU) method. This is called extended NU method for this paper. The eigenvalue solutions of Heun equation and confluent Heun equation are obtained via extended NU method. Some quantum mechanical problems such as Coulomb problem on a 3-sphere, two Coulombically repelling electrons on a sphere, and hyperbolic double-well potential are investigated by this method.

Overview of Krylov subspace methods with applications to control problems

NASA Technical Reports Server (NTRS)

Saad, Youcef

1989-01-01

An overview of projection methods based on Krylov subspaces are given with emphasis on their application to solving matrix equations that arise in control problems. The main idea of Krylov subspace methods is to generate a basis of the Krylov subspace Span and seek an approximate solution the the original problem from this subspace. Thus, the original matrix problem of size N is approximated by one of dimension m typically much smaller than N. Krylov subspace methods have been very successful in solving linear systems and eigenvalue problems and are now just becoming popular for solving nonlinear equations. It is shown how they can be used to solve partial pole placement problems, Sylvester's equation, and Lyapunov's equation.

Analytical solution for the advection-dispersion transport equation in layered media

USDA-ARS?s Scientific Manuscript database

The advection-dispersion transport equation with first-order decay was solved analytically for multi-layered media using the classic integral transform technique (CITT). The solution procedure used an associated non-self-adjoint advection-diffusion eigenvalue problem that had the same form and coef...

Solving the transport equation with quadratic finite elements: Theory and applications

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ferguson, J.M.

1997-12-31

At the 4th Joint Conference on Computational Mathematics, the author presented a paper introducing a new quadratic finite element scheme (QFEM) for solving the transport equation. In the ensuing year the author has obtained considerable experience in the application of this method, including solution of eigenvalue problems, transmission problems, and solution of the adjoint form of the equation as well as the usual forward solution. He will present detailed results, and will also discuss other refinements of his transport codes, particularly for 3-dimensional problems on rectilinear and non-rectilinear grids.

Convergence of the Light-Front Coupled-Cluster Method in Scalar Yukawa Theory

NASA Astrophysics Data System (ADS)

Usselman, Austin

We use Fock-state expansions and the Light-Front Coupled-Cluster (LFCC) method to study mass eigenvalue problems in quantum field theory. Specifically, we study convergence of the method in scalar Yukawa theory. In this theory, a single charged particle is surrounded by a cloud of neutral particles. The charged particle can create or annihilate neutral particles, causing the n-particle state to depend on the n + 1 and n - 1-particle state. Fock state expansion leads to an infinite set of coupled equations where truncation is required. The wave functions for the particle states are expanded in a basis of symmetric polynomials and a generalized eigenvalue problem is solved for the mass eigenvalue. The mass eigenvalue problem is solved for multiple values for the coupling strength while the number of particle states and polynomial basis order are increased. Convergence of the mass eigenvalue solutions is then obtained. Three mass ratios between the charged particle and neutral particles were studied. This includes a massive charged particle, equal masses and massive neutral particles. Relative probability between states can also be explored for more detailed understanding of the process of convergence with respect to the number of Fock sectors. The reliance on higher order particle states depended on how large the mass of the charge particle was. The higher the mass of the charged particle, the more the system depended on higher order particle states. The LFCC method solves this same mass eigenvalue problem using an exponential operator. This exponential operator can then be truncated instead to form a finite system of equations that can be solved using a built in system solver provided in most computational environments, such as MatLab and Mathematica. First approximation in the LFCC method allows for only one particle to be created by the new operator and proved to be not powerful enough to match the Fock state expansion. The second order approximation allowed one and two particles to be created by the new operator and converged to the Fock state expansion results. This showed the LFCC method to be a reliable replacement method for solving quantum field theory problems.

Expendable launch vehicle studies

NASA Technical Reports Server (NTRS)

Bainum, Peter M.; Reiss, Robert

1995-01-01

Analytical support studies of expendable launch vehicles concentrate on the stability of the dynamics during launch especially during or near the region of maximum dynamic pressure. The in-plane dynamic equations of a generic launch vehicle with multiple flexible bending and fuel sloshing modes are developed and linearized. The information from LeRC about the grids, masses, and modes is incorporated into the model. The eigenvalues of the plant are analyzed for several modeling factors: utilizing diagonal mass matrix, uniform beam assumption, inclusion of aerodynamics, and the interaction between the aerodynamics and the flexible bending motion. Preliminary PID, LQR, and LQG control designs with sensor and actuator dynamics for this system and simulations are also conducted. The initial analysis for comparison of PD (proportional-derivative) and full state feedback LQR Linear quadratic regulator) shows that the split weighted LQR controller has better performance than that of the PD. In order to meet both the performance and robustness requirements, the H(sub infinity) robust controller for the expendable launch vehicle is developed. The simulation indicates that both the performance and robustness of the H(sub infinity) controller are better than that for the PID and LQG controllers. The modelling and analysis support studies team has continued development of methodology, using eigensensitivity analysis, to solve three classes of discrete eigenvalue equations. In the first class, the matrix elements are non-linear functions of the eigenvector. All non-linear periodic motion can be cast in this form. Here the eigenvector is comprised of the coefficients of complete basis functions spanning the response space and the eigenvalue is the frequency. The second class of eigenvalue problems studied is the quadratic eigenvalue problem. Solutions for linear viscously damped structures or viscoelastic structures can be reduced to this form. Particular attention is paid to Maxwell and Kelvin models. The third class of problems consists of linear eigenvalue problems in which the elements of the mass and stiffness matrices are stochastic. dynamic structural response for which the parameters are given by probabilistic distribution functions, rather than deterministic values, can be cast in this form. Solutions for several problems in each class will be presented.

The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

NASA Technical Reports Server (NTRS)

Beam, Richard M.; Warming, Robert F.

1991-01-01

Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations.

Introduction to Numerical Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schoonover, Joseph A.

2016-06-14

These are slides for a lecture for the Parallel Computing Summer Research Internship at the National Security Education Center. This gives an introduction to numerical methods. Repetitive algorithms are used to obtain approximate solutions to mathematical problems, using sorting, searching, root finding, optimization, interpolation, extrapolation, least squares regresion, Eigenvalue problems, ordinary differential equations, and partial differential equations. Many equations are shown. Discretizations allow us to approximate solutions to mathematical models of physical systems using a repetitive algorithm and introduce errors that can lead to numerical instabilities if we are not careful.

Lasing eigenvalue problems: the electromagnetic modelling of microlasers

NASA Astrophysics Data System (ADS)

Benson, Trevor; Nosich, Alexander; Smotrova, Elena; Balaban, Mikhail; Sewell, Phillip

2007-02-01

Comprehensive microcavity laser models should account for several physical mechanisms, e.g. carrier transport, heating and optical confinement, coupled by non-linear effects. Nevertheless, considerable useful information can still be obtained if all non-electromagnetic effects are neglected, often within an additional effective-index reduction to an equivalent 2D problem, and the optical modes viewed as solutions of Maxwell's equations. Integral equation (IE) formulations have many advantages over numerical techniques such as FDTD for the study of such microcavity laser problems. The most notable advantages of an IE approach are computational efficiency, the correct description of cavity boundaries without stair-step errors, and the direct solution of an eigenvalue problem rather than the spectral analysis of a transient signal. Boundary IE (BIE) formulations are more economic that volume IE (VIE) ones, because of their lower dimensionality, but they are only applicable to the constant cavity refractive index case. The Muller BIE, being free of 'defect' frequencies and having smooth or integrable kernels, provides a reliable tool for the modal analysis of microcavities. Whilst such an approach can readily identify complex-valued natural frequencies and Q-factors, the lasing condition is not addressed directly. We have thus suggested using a Muller BIE approach to solve a lasing eigenvalue problem (LEP), i.e. a linear eigenvalue solution in the form of two real-valued numbers (lasing wavelength and threshold information) when macroscopic gain is introduced into the cavity material within an active region. Such an approach yields clear insight into the lasing thresholds of individual cavities with uniform and non-uniform gain, cavities coupled as photonic molecules and cavities equipped with one or more quantum dots.

Acoustic modes in fluid networks

NASA Technical Reports Server (NTRS)

Michalopoulos, C. D.; Clark, Robert W., Jr.; Doiron, Harold H.

1992-01-01

Pressure and flow rate eigenvalue problems for one-dimensional flow of a fluid in a network of pipes are derived from the familiar transmission line equations. These equations are linearized by assuming small velocity and pressure oscillations about mean flow conditions. It is shown that the flow rate eigenvalues are the same as the pressure eigenvalues and the relationship between line pressure modes and flow rate modes is established. A volume at the end of each branch is employed which allows any combination of boundary conditions, from open to closed, to be used. The Jacobi iterative method is used to compute undamped natural frequencies and associated pressure/flow modes. Several numerical examples are presented which include acoustic modes for the Helium Supply System of the Space Shuttle Orbiter Main Propulsion System. It should be noted that the method presented herein can be applied to any one-dimensional acoustic system involving an arbitrary number of branches.

Instability of the cored barotropic disc: the linear eigenvalue formulation

NASA Astrophysics Data System (ADS)

Polyachenko, E. V.

2018-05-01

Gaseous rotating razor-thin discs are a testing ground for theories of spiral structure that try to explain appearance and diversity of disc galaxy patterns. These patterns are believed to arise spontaneously under the action of gravitational instability, but calculations of its characteristics in the gas are mostly obscured. The paper suggests a new method for finding the spiral patterns based on an expansion of small amplitude perturbations over Lagrange polynomials in small radial elements. The final matrix equation is extracted from the original hydrodynamical equations without the use of an approximate theory and has a form of the linear algebraic eigenvalue problem. The method is applied to a galactic model with the cored exponential density profile.

S4 solution of the transport equation for eigenvalues using Legendre polynomials

NASA Astrophysics Data System (ADS)

Öztürk, Hakan; Bülbül, Ahmet

2017-09-01

Numerical solution of the transport equation for monoenergetic neutrons scattered isotropically through the medium of a finite homogeneous slab is studied for the determination of the eigenvalues. After obtaining the discrete ordinates form of the transport equation, separated homogeneous and particular solutions are formed and then the eigenvalues are calculated using the Gauss-Legendre quadrature set. Then, the calculated eigenvalues for various values of the c0, the mean number of secondary neutrons per collision, are given in the tables.

On the linear stability of blood flow through model capillary networks.

PubMed

Davis, Jeffrey M

2014-12-01

Under the approximation that blood behaves as a continuum, a numerical implementation is presented to analyze the linear stability of capillary blood flow through model tree and honeycomb networks that are based on the microvascular structures of biological tissues. The tree network is comprised of a cascade of diverging bifurcations, in which a parent vessel bifurcates into two descendent vessels, while the honeycomb network also contains converging bifurcations, in which two parent vessels merge into one descendent vessel. At diverging bifurcations, a cell partitioning law is required to account for the nonuniform distribution of red blood cells as a function of the flow rate of blood into each descendent vessel. A linearization of the governing equations produces a system of delay differential equations involving the discharge hematocrit entering each network vessel and leads to a nonlinear eigenvalue problem. All eigenvalues in a specified region of the complex plane are captured using a transformation based on contour integrals to construct a linear eigenvalue problem with identical eigenvalues, which are then determined using a standard QR algorithm. The predicted value of the dimensionless exponent in the cell partitioning law at the instability threshold corresponds to a supercritical Hopf bifurcation in numerical simulations of the equations governing unsteady blood flow. Excellent agreement is found between the predictions of the linear stability analysis and nonlinear simulations. The relaxation of the assumption of plug flow made in previous stability analyses typically has a small, quantitative effect on the stability results that depends on the specific network structure. This implementation of the stability analysis can be applied to large networks with arbitrary structure provided only that the connectivity among the network segments is known.

Overview of the ArbiTER edge plasma eigenvalue code

NASA Astrophysics Data System (ADS)

Baver, Derek; Myra, James; Umansky, Maxim

2011-10-01

The Arbitrary Topology Equation Reader, or ArbiTER, is a flexible eigenvalue solver that is currently under development for plasma physics applications. The ArbiTER code builds on the equation parser framework of the existing 2DX code, extending it to include a topology parser. This will give the code the capability to model problems with complicated geometries (such as multiple X-points and scrape-off layers) or model equations with arbitrary numbers of dimensions (e.g. for kinetic analysis). In the equation parser framework, model equations are not included in the program's source code. Instead, an input file contains instructions for building a matrix from profile functions and elementary differential operators. The program then executes these instructions in a sequential manner. These instructions may also be translated into analytic form, thus giving the code transparency as well as flexibility. We will present an overview of how the ArbiTER code is to work, as well as preliminary results from early versions of this code. Work supported by the U.S. DOE.

The construction of partner potential from the general potential anharmonic in D-dimensional Schrodinger system

NASA Astrophysics Data System (ADS)

Suparmi; Cari, C.; Wea, K. N.; Wahyulianti

2018-03-01

The Schrodinger equation is the fundamental equation in quantum physics. The characteristic of the particle in physics potential field can be explained by using the Schrodinger equation. In this study, the solution of 4 dimensional Schrodinger equation for the anharmonic potential and the anharmonic partner potential have done. The method that used to solve the Schrodinger equation was the ansatz wave method, while to construction the partner potential was the supersymmetric method. The construction of partner potential used to explain the experiment result that cannot be explained by the original potential. The eigenvalue for anharmonic potential and the anharmonic partner potential have the same characteristic. Every increase of quantum orbital number the eigenvalue getting smaller. This result corresponds to Bohrn’s atomic theory that the eigenvalue is inversely proportional to the atomic shell. But the eigenvalue for the anharmonic partner potential higher than the eigenvalue for the anharmonic original potential.

Recent advances in computational-analytical integral transforms for convection-diffusion problems

NASA Astrophysics Data System (ADS)

Cotta, R. M.; Naveira-Cotta, C. P.; Knupp, D. C.; Zotin, J. L. Z.; Pontes, P. C.; Almeida, A. P.

2017-10-01

An unifying overview of the Generalized Integral Transform Technique (GITT) as a computational-analytical approach for solving convection-diffusion problems is presented. This work is aimed at bringing together some of the most recent developments on both accuracy and convergence improvements on this well-established hybrid numerical-analytical methodology for partial differential equations. Special emphasis is given to novel algorithm implementations, all directly connected to enhancing the eigenfunction expansion basis, such as a single domain reformulation strategy for handling complex geometries, an integral balance scheme in dealing with multiscale problems, the adoption of convective eigenvalue problems in formulations with significant convection effects, and the direct integral transformation of nonlinear convection-diffusion problems based on nonlinear eigenvalue problems. Then, selected examples are presented that illustrate the improvement achieved in each class of extension, in terms of convergence acceleration and accuracy gain, which are related to conjugated heat transfer in complex or multiscale microchannel-substrate geometries, multidimensional Burgers equation model, and diffusive metal extraction through polymeric hollow fiber membranes. Numerical results are reported for each application and, where appropriate, critically compared against the traditional GITT scheme without convergence enhancement schemes and commercial or dedicated purely numerical approaches.

Two-dimensional integrating matrices on rectangular grids. [solving differential equations associated with rotating structures

NASA Technical Reports Server (NTRS)

Lakin, W. D.

1981-01-01

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

Program for solution of ordinary differential equations

NASA Technical Reports Server (NTRS)

Sloate, H.

1973-01-01

A program for the solution of linear and nonlinear first order ordinary differential equations is described and user instructions are included. The program contains a new integration algorithm for the solution of initial value problems which is particularly efficient for the solution of differential equations with a wide range of eigenvalues. The program in its present form handles up to ten state variables, but expansion to handle up to fifty state variables is being investigated.

Closed-form eigensolutions of nonviscously, nonproportionally damped systems based on continuous damping sensitivity

NASA Astrophysics Data System (ADS)

Lázaro, Mario

2018-01-01

In this paper, nonviscous, nonproportional, vibrating structures are considered. Nonviscously damped systems are characterized by dissipative mechanisms which depend on the history of the response velocities via hereditary kernel functions. Solutions of the free motion equation lead to a nonlinear eigenvalue problem involving mass, stiffness and damping matrices. Viscoelasticity leads to a frequency dependence of this latter. In this work, a novel closed-form expression to estimate complex eigenvalues is derived. The key point is to consider the damping model as perturbed by a continuous fictitious parameter. Assuming then the eigensolutions as function of this parameter, the computation of the eigenvalues sensitivity leads to an ordinary differential equation, from whose solution arises the proposed analytical formula. The resulting expression explicitly depends on the viscoelasticity (frequency derivatives of the damping function), the nonproportionality (influence of the modal damping matrix off-diagonal terms). Eigenvectors are obtained using existing methods requiring only the corresponding eigenvalue. The method is validated using a numerical example which compares proposed with exact ones and with those determined from the linear first order approximation in terms of the damping matrix. Frequency response functions are also plotted showing that the proposed approach is valid even for moderately or highly damped systems.

Asymptotic Representation for the Eigenvalues of a Non-selfadjoint Operator Governing the Dynamics of an Energy Harvesting Model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shubov, Marianna A., E-mail: marianna.shubov@gmail.com

2016-06-15

We consider a well known model of a piezoelectric energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler–Bernoulli model for the transverse vibrations of the beam and the second one represents the Kirchhoff’s law for the electric circuit. Both equations aremore » coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. Our main result is an explicit asymptotic formula for the eigenvalues of this generator, i.e., we perform the modal analysis for electrically loaded (not short-circuit) system. We show that the spectrum splits into an infinite sequence of stable eigenvalues that approaches a vertical line in the left half plane and possibly of a finite number of unstable eigenvalues. This paper is the first in a series of three works. In the second one we will prove that the generalized eigenvectors of the dynamics generator form a Riesz basis (and, moreover, a Bari basis) in the energy space. In the third paper we will apply the results of the first two to control problems for this model.« less

The Guderley problem revisited

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ramsey, Scott D; Kamm, James R; Bolstad, John H

2009-01-01

The self-similar converging-diverging shock wave problem introduced by Guderley in 1942 has been the source of numerous investigations since its publication. In this paper, we review the simplifications and group invariance properties that lead to a self-similar formulation of this problem from the compressible flow equations for a polytropic gas. The complete solution to the self-similar problem reduces to two coupled nonlinear eigenvalue problems: the eigenvalue of the first is the so-called similarity exponent for the converging flow, and that of the second is a trajectory multiplier for the diverging regime. We provide a clear exposition concerning the reflected shockmore » configuration. Additionally, we introduce a new approximation for the similarity exponent, which we compare with other estimates and numerically computed values. Lastly, we use the Guderley problem as the basis of a quantitative verification analysis of a cell-centered, finite volume, Eulerian compressible flow algorithm.« less

Computationally efficient finite-difference modal method for the solution of Maxwell's equations.

PubMed

Semenikhin, Igor; Zanuccoli, Mauro

2013-12-01

In this work, a new implementation of the finite-difference (FD) modal method (FDMM) based on an iterative approach to calculate the eigenvalues and corresponding eigenfunctions of the Helmholtz equation is presented. Two relevant enhancements that significantly increase the speed and accuracy of the method are introduced. First of all, the solution of the complete eigenvalue problem is avoided in favor of finding only the meaningful part of eigenmodes by using iterative methods. Second, a multigrid algorithm and Richardson extrapolation are implemented. Simultaneous use of these techniques leads to an enhancement in terms of accuracy, which allows a simple method such as the FDMM with a typical three-point difference scheme to be significantly competitive with an analytical modal method.

Vibration properties of and power harvested by a system of electromagnetic vibration energy harvesters that have electrical dynamics

NASA Astrophysics Data System (ADS)

Cooley, Christopher G.

2017-09-01

This study investigates the vibration and dynamic response of a system of coupled electromagnetic vibration energy harvesting devices that each consist of a proof mass, elastic structure, electromagnetic generator, and energy harvesting circuit with inductance, resistance, and capacitance. The governing equations for the coupled electromechanical system are derived using Newtonian mechanics and Kirchhoff circuit laws for an arbitrary number of these subsystems. The equations are cast in matrix operator form to expose the device's vibration properties. The device's complex-valued eigenvalues and eigenvectors are related to physical characteristics of its vibration. Because the electrical circuit has dynamics, these devices have more natural frequencies than typical electromagnetic vibration energy harvesters that have purely resistive circuits. Closed-form expressions for the steady state dynamic response and average power harvested are derived for devices with a single subsystem. Example numerical results for single and double subsystem devices show that the natural frequencies and vibration modes obtained from the eigenvalue problem agree with the resonance locations and response amplitudes obtained independently from forced response calculations. This agreement demonstrates the usefulness of solving eigenvalue problems for these devices. The average power harvested by the device differs substantially at each resonance. Devices with multiple subsystems have multiple modes where large amounts of power are harvested.

Linear state feedback, quadratic weights, and closed loop eigenstructures. M.S. Thesis. Final Report

NASA Technical Reports Server (NTRS)

Thompson, P. M.

1980-01-01

Equations are derived for the angles of general multivariable root loci and linear quadratic optimal root loci, including angles of departure and approach. The generalized eigenvalue problem is used to compute angles of approach. Equations are also derived to find the sensitivity of closed loop eigenvalue and the directional derivatives of closed loop eigenvectors. An equivalence class of quadratic weights that produce the same asymptotic eigenstructure is defined, a canonical element is defined, and an algorithm to find it is given. The behavior of the optimal root locus in the nonasymptotic region is shown to be different for quadratic weights with the same asymptotic properties. An algorithm is presented that can be used to select a feedback gain matrix for the linear state feedback problem which produces a specified asymptotic eigenstructure. Another algorithm is given to compute the asymptotic eigenstructure properties inherent in a given set of quadratic weights. Finally, it is shown that optimal root loci for nongeneric problems can be approximated by generic ones in the nonasymptotic region.

Linear state feedback, quadratic weights, and closed loop eigenstructures. M.S. Thesis

NASA Technical Reports Server (NTRS)

Thompson, P. M.

1979-01-01

Results are given on the relationships between closed loop eigenstructures, state feedback gain matrices of the linear state feedback problem, and quadratic weights of the linear quadratic regulator. Equations are derived for the angles of general multivariable root loci and linear quadratic optimal root loci, including angles of departure and approach. The generalized eigenvalue problem is used for the first time to compute angles of approach. Equations are also derived to find the sensitivity of closed loop eigenvalues and the directional derivatives of closed loop eigenvectors (with respect to a scalar multiplying the feedback gain matrix or the quadratic control weight). An equivalence class of quadratic weights that produce the same asymptotic eigenstructure is defined, sufficient conditions to be in it are given, a canonical element is defined, and an algorithm to find it is given. The behavior of the optimal root locus in the nonasymptotic region is shown to be different for quadratic weights with the same asymptotic properties.

Fourier analysis of finite element preconditioned collocation schemes

NASA Technical Reports Server (NTRS)

Deville, Michel O.; Mund, Ernest H.

1990-01-01

The spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes is investigated. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advection-diffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential equations resulting from Fourier analysis (in the tranverse direction) of the 2-D Stokes problem. All results agree with previous conclusions on the numerical efficiency of finite element preconditioning schemes.

Exact dark soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media.

PubMed

Nakkeeran, K

2001-10-01

We consider a family of N coupled nonlinear Schrödinger equations which govern the simultaneous propagation of N fields in the normal dispersion regime of an optical fiber with various important physical effects. The linear eigenvalue problem associated with the integrable form of all the equations is constructed with the help of the Ablowitz-Kaup-Newell-Segur method. Using the Hirota bilinear method, exact dark soliton solutions are explicitly derived.

Ghost instabilities of cosmological models with vector fields nonminimally coupled to the curvature

DOE Office of Scientific and Technical Information (OSTI.GOV)

Himmetoglu, Burak; Peloso, Marco; Contaldi, Carlo R.

2009-12-15

We prove that many cosmological models characterized by vectors nonminimally coupled to the curvature (such as the Turner-Widrow mechanism for the production of magnetic fields during inflation, and models of vector inflation or vector curvaton) contain ghosts. The ghosts are associated with the longitudinal vector polarization present in these models and are found from studying the sign of the eigenvalues of the kinetic matrix for the physical perturbations. Ghosts introduce two main problems: (1) they make the theories ill defined at the quantum level in the high energy/subhorizon regime (and create serious problems for finding a well-behaved UV completion), andmore » (2) they create an instability already at the linearized level. This happens because the eigenvalue corresponding to the ghost crosses zero during the cosmological evolution. At this point the linearized equations for the perturbations become singular (we show that this happens for all the models mentioned above). We explicitly solve the equations in the simplest cases of a vector without a vacuum expectation value in a Friedmann-Robertson-Walker geometry, and of a vector with a vacuum expectation value plus a cosmological constant, and we show that indeed the solutions of the linearized equations diverge when these equations become singular.« less

Eigenvalue sensitivity analysis of planar frames with variable joint and support locations

NASA Technical Reports Server (NTRS)

Chuang, Ching H.; Hou, Gene J. W.

1991-01-01

Two sensitivity equations are derived in this study based upon the continuum approach for eigenvalue sensitivity analysis of planar frame structures with variable joint and support locations. A variational form of an eigenvalue equation is first derived in which all of the quantities are expressed in the local coordinate system attached to each member. Material derivative of this variational equation is then sought to account for changes in member's length and orientation resulting form the perturbation of joint and support locations. Finally, eigenvalue sensitivity equations are formulated in either domain quantities (by the domain method) or boundary quantities (by the boundary method). It is concluded that the sensitivity equation derived by the boundary method is more efficient in computation but less accurate than that of the domain method. Nevertheless, both of them in terms of computational efficiency are superior to the conventional direct differentiation method and the finite difference method.

An accurate and efficient acoustic eigensolver based on a fast multipole BEM and a contour integral method

NASA Astrophysics Data System (ADS)

Zheng, Chang-Jun; Gao, Hai-Feng; Du, Lei; Chen, Hai-Bo; Zhang, Chuanzeng

2016-01-01

An accurate numerical solver is developed in this paper for eigenproblems governed by the Helmholtz equation and formulated through the boundary element method. A contour integral method is used to convert the nonlinear eigenproblem into an ordinary eigenproblem, so that eigenvalues can be extracted accurately by solving a set of standard boundary element systems of equations. In order to accelerate the solution procedure, the parameters affecting the accuracy and efficiency of the method are studied and two contour paths are compared. Moreover, a wideband fast multipole method is implemented with a block IDR (s) solver to reduce the overall solution cost of the boundary element systems of equations with multiple right-hand sides. The Burton-Miller formulation is employed to identify the fictitious eigenfrequencies of the interior acoustic problems with multiply connected domains. The actual effect of the Burton-Miller formulation on tackling the fictitious eigenfrequency problem is investigated and the optimal choice of the coupling parameter as α = i / k is confirmed through exterior sphere examples. Furthermore, the numerical eigenvalues obtained by the developed method are compared with the results obtained by the finite element method to show the accuracy and efficiency of the developed method.

A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems

NASA Astrophysics Data System (ADS)

Babaee, Hessam; Choi, Minseok; Sapsis, Themistoklis P.; Karniadakis, George Em

2017-09-01

We develop a new robust methodology for the stochastic Navier-Stokes equations based on the dynamically-orthogonal (DO) and bi-orthogonal (BO) methods [1-3]. Both approaches are variants of a generalized Karhunen-Loève (KL) expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics, hence, capturing the low-dimensional structure of the solution. The DO and BO formulations are mathematically equivalent [3], but they exhibit computationally complimentary properties. Specifically, the BO formulation may fail due to crossing of the eigenvalues of the covariance matrix, while both BO and DO become unstable when there is a high condition number of the covariance matrix or zero eigenvalues. To this end, we combine the two methods into a robust hybrid framework and in addition we employ a pseudo-inverse technique to invert the covariance matrix. The robustness of the proposed method stems from addressing the following issues in the DO/BO formulation: (i) eigenvalue crossing: we resolve the issue of eigenvalue crossing in the BO formulation by switching to the DO near eigenvalue crossing using the equivalence theorem and switching back to BO when the distance between eigenvalues is larger than a threshold value; (ii) ill-conditioned covariance matrix: we utilize a pseudo-inverse strategy to invert the covariance matrix; (iii) adaptivity: we utilize an adaptive strategy to add/remove modes to resolve the covariance matrix up to a threshold value. In particular, we introduce a soft-threshold criterion to allow the system to adapt to the newly added/removed mode and therefore avoid repetitive and unnecessary mode addition/removal. When the total variance approaches zero, we show that the DO/BO formulation becomes equivalent to the evolution equation of the Optimally Time-Dependent modes [4]. We demonstrate the capability of the proposed methodology with several numerical examples, namely (i) stochastic Burgers equation: we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues; (ii) stochastic Kovasznay flow: we examine the method in the presence of a singular covariance matrix; and (iii) we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen DO/BO modes are active.

Hierarchical coarse-graining transform.

PubMed

Pancaldi, Vera; King, Peter R; Christensen, Kim

2009-03-01

We present a hierarchical transform that can be applied to Laplace-like differential equations such as Darcy's equation for single-phase flow in a porous medium. A finite-difference discretization scheme is used to set the equation in the form of an eigenvalue problem. Within the formalism suggested, the pressure field is decomposed into an average value and fluctuations of different kinds and at different scales. The application of the transform to the equation allows us to calculate the unknown pressure with a varying level of detail. A procedure is suggested to localize important features in the pressure field based only on the fine-scale permeability, and hence we develop a form of adaptive coarse graining. The formalism and method are described and demonstrated using two synthetic toy problems.

The Coulomb problem on a 3-sphere and Heun polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bellucci, Stefano; Yeghikyan, Vahagn; Yerevan State University, Alex-Manoogian st. 1, 00025 Yerevan

2013-08-15

The paper studies the quantum mechanical Coulomb problem on a 3-sphere. We present a special parametrization of the ellipto-spheroidal coordinate system suitable for the separation of variables. After quantization we get the explicit form of the spectrum and present an algebraic equation for the eigenvalues of the Runge-Lentz vector. We also present the wave functions expressed via Heun polynomials.

Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem

NASA Technical Reports Server (NTRS)

Ceccobello, C.; Farinelli, R.; Titarchuk, L.

2014-01-01

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

A Riemann solver for RANS

NASA Astrophysics Data System (ADS)

Chuvakhov, P. V.

2014-01-01

An exact expression for a system of both eigenvalues and right/left eigenvectors of a Jacobian matrix for a convective two-equation differential closure RANS operator split along a curvilinear coordinate is derived. It is shown by examples of numerical modeling of supersonic flows over a flat plate and a compression corner with separation that application of the exact system of eigenvalues and eigenvectors to the Roe approach for approximate solution of the Riemann problem gives rise to an increase in the convergence rate, better stability and higher accuracy of a steady-state solution in comparison with those in the case of an approximate system.

Double layers without current

DOE Office of Scientific and Technical Information (OSTI.GOV)

Perkins, F.W.; Sun, Y.C.

1980-11-01

The steady-state solution of the nonlinear Vlasov-Poisson equations is reduced to a nonlinear eigenvalue problem for the case of double-layer (potential drop) boundary conditions. Solutions with no relative electron-ion drifts are found. The kinetic stability is discussed. Suggestions for creating these states in experiments and computer simulations are offered.

Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

NASA Astrophysics Data System (ADS)

Huang, Tsung-Ming; Lin, Wen-Wei; Tian, Heng; Chen, Guan-Hua

2018-03-01

Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000 × 4000 at the density functional tight binding level, corresponding to a 8 × 8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

Matrix eigenvalue method for free-oscillations modelling of spherical elastic bodies

NASA Astrophysics Data System (ADS)

Zábranová, E.; Hanyk, L.; Matyska, C.

2017-11-01

Deformations and changes of the gravitational potential of pre-stressed self-gravitating elastic bodies caused by free oscillations are described by means of the momentum and Poisson equations and the constitutive relation. For spherically symmetric bodies, the equations and boundary conditions are transformed into ordinary differential equations of the second order by the spherical harmonic decomposition and further discretized by highly accurate pseudospectral difference schemes on Chebyshev grids; we pay special attention to the conditions at the centre of the models. We thus obtain a series of matrix eigenvalue problems for eigenfrequencies and eigenfunctions of the free oscillations. Accuracy of the presented numerical approach is tested by means of the Rayleigh quotients calculated for the eigenfrequencies up to 500 mHz. Both the modal frequencies and eigenfunctions are benchmarked against the output from the Mineos software package based on shooting methods. The presented technique is a promising alternative to widely used methods because it is stable and with a good capability up to high frequencies.

Matrix Sturm-Liouville equation with a Bessel-type singularity on a finite interval

NASA Astrophysics Data System (ADS)

Bondarenko, Natalia

2017-03-01

The matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with the prescribed behavior at the singular point and Birkhoff-type solutions with the known asymptotics for large values of the spectral parameter. The asymptotic formulas for Stokes multipliers, connecting these two fundamental systems of solutions, are derived. We also set boundary conditions and obtain asymptotic formulas for the spectral data (the eigenvalues and the weight matrices) of the boundary value problem. Our results will be useful in the theory of direct and inverse spectral problems.

Constrained multibody system dynamics: An automated approach

NASA Technical Reports Server (NTRS)

Kamman, J. W.; Huston, R. L.

1982-01-01

The governing equations for constrained multibody systems are formulated in a manner suitable for their automated, numerical development and solution. The closed loop problem of multibody chain systems is addressed. The governing equations are developed by modifying dynamical equations obtained from Lagrange's form of d'Alembert's principle. The modifications is based upon a solution of the constraint equations obtained through a zero eigenvalues theorem, is a contraction of the dynamical equations. For a system with n-generalized coordinates and m-constraint equations, the coefficients in the constraint equations may be viewed as constraint vectors in n-dimensional space. In this setting the system itself is free to move in the n-m directions which are orthogonal to the constraint vectors.

Application of the Finite Element Method to Rotary Wing Aeroelasticity

NASA Technical Reports Server (NTRS)

Straub, F. K.; Friedmann, P. P.

1982-01-01

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method.

On Instability of Geostrophic Current with Linear Vertical Shear at Length Scales of Interleaving

NASA Astrophysics Data System (ADS)

Kuzmina, N. P.; Skorokhodov, S. L.; Zhurbas, N. V.; Lyzhkov, D. A.

2018-01-01

The instability of long-wave disturbances of a geostrophic current with linear velocity shear is studied with allowance for the diffusion of buoyancy. A detailed derivation of the model problem in dimensionless variables is presented, which is used for analyzing the dynamics of disturbances in a vertically bounded layer and for describing the formation of large-scale intrusions in the Arctic basin. The problem is solved numerically based on a high-precision method developed for solving fourth-order differential equations. It is established that there is an eigenvalue in the spectrum of eigenvalues that corresponds to unstable (growing with time) disturbances, which are characterized by a phase velocity exceeding the maximum velocity of the geostrophic flow. A discussion is presented to explain some features of the instability.

Graph theory approach to the eigenvalue problem of large space structures

NASA Technical Reports Server (NTRS)

Reddy, A. S. S. R.; Bainum, P. M.

1981-01-01

Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.

Guided waves dispersion equations for orthotropic multilayered pipes solved using standard finite elements code.

PubMed

Predoi, Mihai Valentin

2014-09-01

The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0,n), torsional T(0,n) and flexural F(m,n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method. Copyright © 2014 Elsevier B.V. All rights reserved.

Algorithms and software for solving finite element equations on serial and parallel architectures

NASA Technical Reports Server (NTRS)

George, Alan

1989-01-01

Over the past 15 years numerous new techniques have been developed for solving systems of equations and eigenvalue problems arising in finite element computations. A package called SPARSPAK has been developed by the author and his co-workers which exploits these new methods. The broad objective of this research project is to incorporate some of this software in the Computational Structural Mechanics (CSM) testbed, and to extend the techniques for use on multiprocessor architectures.

Aeroelastic analysis of a troposkien-type wind turbine blade

NASA Technical Reports Server (NTRS)

Nitzsche, F.

1981-01-01

The linear aeroelastic equations for one curved blade of a vertical axis wind turbine in state vector form are presented. The method is based on a simple integrating matrix scheme together with the transfer matrix idea. The method is proposed as a convenient way of solving the associated eigenvalue problem for general support conditions.

A fixed energy fixed angle inverse scattering in interior transmission problem

NASA Astrophysics Data System (ADS)

Chen, Lung-Hui

2017-06-01

We study the inverse acoustic scattering problem in mathematical physics. The problem is to recover the index of refraction in an inhomogeneous medium by measuring the scattered wave fields in the far field. We transform the problem to the interior transmission problem in the study of the Helmholtz equation. We find an inverse uniqueness on the scatterer with a knowledge of a fixed interior transmission eigenvalue. By examining the solution in a series of spherical harmonics in the far field, we can determine uniquely the perturbation source for the radially symmetric perturbations.

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

NASA Technical Reports Server (NTRS)

Kennedy, Christopher A.; Carpenter, Mark H.

2016-01-01

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.

A spectral approach for the stability analysis of turbulent open-channel flows over granular beds

NASA Astrophysics Data System (ADS)

Camporeale, C.; Canuto, C.; Ridolfi, L.

2012-01-01

A novel Orr-Sommerfeld-like equation for gravity-driven turbulent open-channel flows over a granular erodible bed is here derived, and the linear stability analysis is developed. The whole spectrum of eigenvalues and eigenvectors of the complete generalized eigenvalue problem is computed and analyzed. The fourth-order eigenvalue problem presents singular non-polynomial coefficients with non-homogenous Robin-type boundary conditions that involve first and second derivatives. Furthermore, the Exner condition is imposed at an internal point. We propose a numerical discretization of spectral type based on a single-domain Galerkin scheme. In order to manage the presence of singular coefficients, some properties of Jacobi polynomials have been carefully blended with numerical integration of Gauss-Legendre type. The results show a positive agreement with the classical experimental data and allow one to relate the different types of instability to such parameters as the Froude number, wavenumber, and the roughness scale. The eigenfunctions allow two types of boundary layers to be distinguished, scaling, respectively, with the roughness height and the saltation layer for the bedload sediment transport.

An assessment of coupling algorithms for nuclear reactor core physics simulations

DOE PAGES

Hamilton, Steven; Berrill, Mark; Clarno, Kevin; ...

2016-04-01

This paper evaluates the performance of multiphysics coupling algorithms applied to a light water nuclear reactor core simulation. The simulation couples the k-eigenvalue form of the neutron transport equation with heat conduction and subchannel flow equations. We compare Picard iteration (block Gauss–Seidel) to Anderson acceleration and multiple variants of preconditioned Jacobian-free Newton–Krylov (JFNK). The performance of the methods are evaluated over a range of energy group structures and core power levels. A novel physics-based approximation to a Jacobian-vector product has been developed to mitigate the impact of expensive on-line cross section processing steps. Furthermore, numerical simulations demonstrating the efficiency ofmore » JFNK and Anderson acceleration relative to standard Picard iteration are performed on a 3D model of a nuclear fuel assembly. Both criticality (k-eigenvalue) and critical boron search problems are considered.« less

An assessment of coupling algorithms for nuclear reactor core physics simulations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hamilton, Steven; Berrill, Mark; Clarno, Kevin

This paper evaluates the performance of multiphysics coupling algorithms applied to a light water nuclear reactor core simulation. The simulation couples the k-eigenvalue form of the neutron transport equation with heat conduction and subchannel flow equations. We compare Picard iteration (block Gauss–Seidel) to Anderson acceleration and multiple variants of preconditioned Jacobian-free Newton–Krylov (JFNK). The performance of the methods are evaluated over a range of energy group structures and core power levels. A novel physics-based approximation to a Jacobian-vector product has been developed to mitigate the impact of expensive on-line cross section processing steps. Furthermore, numerical simulations demonstrating the efficiency ofmore » JFNK and Anderson acceleration relative to standard Picard iteration are performed on a 3D model of a nuclear fuel assembly. Both criticality (k-eigenvalue) and critical boron search problems are considered.« less

An assessment of coupling algorithms for nuclear reactor core physics simulations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hamilton, Steven, E-mail: hamiltonsp@ornl.gov; Berrill, Mark, E-mail: berrillma@ornl.gov; Clarno, Kevin, E-mail: clarnokt@ornl.gov

This paper evaluates the performance of multiphysics coupling algorithms applied to a light water nuclear reactor core simulation. The simulation couples the k-eigenvalue form of the neutron transport equation with heat conduction and subchannel flow equations. We compare Picard iteration (block Gauss–Seidel) to Anderson acceleration and multiple variants of preconditioned Jacobian-free Newton–Krylov (JFNK). The performance of the methods are evaluated over a range of energy group structures and core power levels. A novel physics-based approximation to a Jacobian-vector product has been developed to mitigate the impact of expensive on-line cross section processing steps. Numerical simulations demonstrating the efficiency of JFNKmore » and Anderson acceleration relative to standard Picard iteration are performed on a 3D model of a nuclear fuel assembly. Both criticality (k-eigenvalue) and critical boron search problems are considered.« less

Tube wave signatures in cylindrically layered poroelastic media computed with spectral method

NASA Astrophysics Data System (ADS)

Karpfinger, Florian; Gurevich, Boris; Valero, Henri-Pierre; Bakulin, Andrey; Sinha, Bikash

2010-11-01

This paper describes a new algorithm based on the spectral method for the computation of Stoneley wave dispersion and attenuation propagating in cylindrical structures composed of fluid, elastic and poroelastic layers. The spectral method is a numerical method which requires discretization of the structure along the radial axis using Chebyshev points. To approximate the differential operators of the underlying differential equations, we use spectral differentiation matrices. After discretizing equations of motion along the radial direction, we can solve the problem as a generalized algebraic eigenvalue problem. For a given frequency, calculated eigenvalues correspond to the wavenumbers of different modes. The advantage of this approach is that it can very efficiently analyse structures with complicated radial layering composed of different fluid, solid and poroelastic layers. This work summarizes the fundamental equations, followed by an outline of how they are implemented in the numerical spectral schema. The interface boundary conditions are then explained for fluid/porous, elastic/porous and porous interfaces. Finally, we discuss three examples from borehole acoustics. The first model is a fluid-filled borehole surrounded by a poroelastic formation. The second considers an additional elastic layer sandwiched between the borehole and the formation, and finally a model with radially increasing permeability is considered.

The eigenvalue spectrum of the Orr-Sommerfeld problem

NASA Technical Reports Server (NTRS)

Antar, B. N.

1976-01-01

A numerical investigation of the temporal eigenvalue spectrum of the ORR-Sommerfeld equation is presented. Two flow profiles are studied, the plane Poiseuille flow profile and the Blasius boundary layer (parallel): flow profile. In both cases a portion of the complex c-plane bounded by 0 less than or equal to CR sub r 1 and -1 less than or equal to ci sub i 0 is searched and the eigenvalues within it are identified. The spectra for the plane Poiseuille flow at alpha = 1.0 and R = 100, 1000, 6000, and 10000 are determined and compared with existing results where possible. The spectrum for the Blasius boundary layer flow at alpha = 0.308 and R = 998 was found to be infinite and discrete. Other spectra for the Blasius boundary layer at various Reynolds numbers seem to confirm this result. The eigenmodes belonging to these spectra were located and discussed.

Krein signature for instability of PT-symmetric states

NASA Astrophysics Data System (ADS)

Chernyavsky, Alexander; Pelinovsky, Dmitry E.

2018-05-01

Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT-symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.

Quantum spatial propagation of squeezed light in a degenerate parametric amplifier

NASA Technical Reports Server (NTRS)

Deutsch, Ivan H.; Garrison, John C.

1992-01-01

Differential equations which describe the steady state spatial evolution of nonclassical light are established using standard quantum field theoretic techniques. A Schroedinger equation for the state vector of the optical field is derived using the quantum analog of the slowly varying envelope approximation (SVEA). The steady state solutions are those that satisfy the time independent Schroedinger equation. The resulting eigenvalue problem then leads to the spatial propagation equations. For the degenerate parametric amplifier this method shows that the squeezing parameter obey nonlinear differential equations coupled by the amplifier gain and phase mismatch. The solution to these differential equations is equivalent to one obtained from the classical three wave mixing steady state solution to the parametric amplifier with a nondepleted pump.

On numerical solutions to the QCD ’t Hooft equation in the limit of large quark mass

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zubov, Roman; Prokhvatilov, Evgeni

2016-01-22

First we give a short informal introduction to the theory behind the ’t Hooft equation. Then we consider numerical solutions to this equation in the limit of large fermion masses. It turns out that the spectrum of eigenvalues coincides with that of the Airy differential equation. Moreover when we take the Fourier transform of eigenfunctions, they look like the corresponding Airy functions with appropriate symmetry. It is known that these functions correspond to solutions of a one dimensional Schrodinger equation for a particle in a triangular potential well. So we find the analogy between this problem and the ’t Hooftmore » equation. We also present a simple intuition behind these results.« less

Parallel-vector computation for structural analysis and nonlinear unconstrained optimization problems

NASA Technical Reports Server (NTRS)

Nguyen, Duc T.

1990-01-01

Practical engineering application can often be formulated in the form of a constrained optimization problem. There are several solution algorithms for solving a constrained optimization problem. One approach is to convert a constrained problem into a series of unconstrained problems. Furthermore, unconstrained solution algorithms can be used as part of the constrained solution algorithms. Structural optimization is an iterative process where one starts with an initial design, a finite element structure analysis is then performed to calculate the response of the system (such as displacements, stresses, eigenvalues, etc.). Based upon the sensitivity information on the objective and constraint functions, an optimizer such as ADS or IDESIGN, can be used to find the new, improved design. For the structural analysis phase, the equation solver for the system of simultaneous, linear equations plays a key role since it is needed for either static, or eigenvalue, or dynamic analysis. For practical, large-scale structural analysis-synthesis applications, computational time can be excessively large. Thus, it is necessary to have a new structural analysis-synthesis code which employs new solution algorithms to exploit both parallel and vector capabilities offered by modern, high performance computers such as the Convex, Cray-2 and Cray-YMP computers. The objective of this research project is, therefore, to incorporate the latest development in the parallel-vector equation solver, PVSOLVE into the widely popular finite-element production code, such as the SAP-4. Furthermore, several nonlinear unconstrained optimization subroutines have also been developed and tested under a parallel computer environment. The unconstrained optimization subroutines are not only useful in their own right, but they can also be incorporated into a more popular constrained optimization code, such as ADS.

Quantum mechanics on space with SU(2) fuzziness

NASA Astrophysics Data System (ADS)

Fatollahi, Amir H.; Shariati, Ahmad; Khorrami, Mohammad

2009-04-01

Quantum mechanics of models is considered which are constructed in spaces with Lie algebra type commutation relations between spatial coordinates. The case is specialized to that of the group SU(2), for which the formulation of the problem via the Euler parameterization is also presented. SU(2)-invariant systems are discussed, and the corresponding eigenvalue problem for the Hamiltonian is reduced to an ordinary differential equation, as is the case with such models on commutative spaces.

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1984-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, Kazufumi; Teglas, Russell

1987-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Optical systolic solutions of linear algebraic equations

NASA Technical Reports Server (NTRS)

Neuman, C. P.; Casasent, D.

1984-01-01

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Artificial neural network methods in quantum mechanics

NASA Astrophysics Data System (ADS)

Lagaris, I. E.; Likas, A.; Fotiadis, D. I.

1997-08-01

In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schrödinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schrödinger and the Dirac equations for a muonic atom, as well as with a nonlocal Schrödinger integrodifferential equation that models the n + α system in the framework of the resonating group method. In two dimensions we consider the well-studied Henon-Heiles Hamiltonian and in three dimensions the model problem of three coupled anharmonic oscillators. The method in all of the treated cases proved to be highly accurate, robust and efficient. Hence it is a promising tool for tackling problems of higher complexity and dimensionality.

Computing singularities of perturbation series

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kvaal, Simen; Jarlebring, Elias; Michiels, Wim

2011-03-15

Many properties of current ab initio approaches to the quantum many-body problem, both perturbational and otherwise, are related to the singularity structure of the Rayleigh-Schroedinger perturbation series. A numerical procedure is presented that in principle computes the complete set of singularities, including the dominant singularity which limits the radius of convergence. The method approximates the singularities as eigenvalues of a certain generalized eigenvalue equation which is solved using iterative techniques. It relies on computation of the action of the Hamiltonian matrix on a vector and does not rely on the terms in the perturbation series. The method can be usefulmore » for studying perturbation series of typical systems of moderate size, for fundamental development of resummation schemes, and for understanding the structure of singularities for typical systems. Some illustrative model problems are studied, including a helium-like model with {delta}-function interactions for which Moeller-Plesset perturbation theory is considered and the radius of convergence found.« less

Twisting singular solutions of Betheʼs equations

NASA Astrophysics Data System (ADS)

Nepomechie, Rafael I.; Wang, Chunguang

2014-12-01

The Bethe equations for the periodic XXX and XXZ spin chains admit singular solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to derive conditions for such singular solutions to be physical, in which case they correspond to genuine eigenvalues and eigenvectors of the Hamiltonian.

Computing Evans functions numerically via boundary-value problems

NASA Astrophysics Data System (ADS)

Barker, Blake; Nguyen, Rose; Sandstede, Björn; Ventura, Nathaniel; Wahl, Colin

2018-03-01

The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.

Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model.

PubMed

Semenov, Yuri S; Novozhilov, Artem S

2015-08-01

We reformulate the eigenvalue problem for the selection-mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples; theoretical findings are compared with numerical calculations. Copyright © 2015. Published by Elsevier Inc.

Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability

NASA Technical Reports Server (NTRS)

Ito, K.

1984-01-01

The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

Variables separation and superintegrability of the nine-dimensional MICZ-Kepler problem

NASA Astrophysics Data System (ADS)

Phan, Ngoc-Hung; Le, Dai-Nam; Thoi, Tuan-Quoc N.; Le, Van-Hoang

2018-03-01

The nine-dimensional MICZ-Kepler problem is of recent interest. This is a system describing a charged particle moving in the Coulomb field plus the field of a SO(8) monopole in a nine-dimensional space. Interestingly, this problem is equivalent to a 16-dimensional harmonic oscillator via the Hurwitz transformation. In the present paper, we report on the multiseparability, a common property of superintegrable systems, and the superintegrability of the problem. First, we show the solvability of the Schrödinger equation of the problem by the variables separation method in different coordinates. Second, based on the SO(10) symmetry algebra of the system, we construct explicitly a set of seventeen invariant operators, which are all in the second order of the momentum components, satisfying the condition of superintegrability. The found number 17 coincides with the prediction of (2n - 1) law of maximal superintegrability order in the case n = 9. Until now, this law is accepted to apply only to scalar Hamiltonian eigenvalue equations in n-dimensional space; therefore, our results can be treated as evidence that this definition of superintegrability may also apply to some vector equations such as the Schrödinger equation for the nine-dimensional MICZ-Kepler problem.

Comparison of two Galerkin quadrature methods

DOE PAGES

Morel, Jim E.; Warsa, James; Franke, Brian C.; ...

2017-02-21

Here, we compare two methods for generating Galerkin quadratures. In method 1, the standard S N method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard S N method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwisemore » sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding S N equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.« less

Efficient parallel resolution of the simplified transport equations in mixed-dual formulation

NASA Astrophysics Data System (ADS)

Barrault, M.; Lathuilière, B.; Ramet, P.; Roman, J.

2011-03-01

A reactivity computation consists of computing the highest eigenvalue of a generalized eigenvalue problem, for which an inverse power algorithm is commonly used. Very fine modelizations are difficult to treat for our sequential solver, based on the simplified transport equations, in terms of memory consumption and computational time. A first implementation of a Lagrangian based domain decomposition method brings to a poor parallel efficiency because of an increase in the power iterations [1]. In order to obtain a high parallel efficiency, we improve the parallelization scheme by changing the location of the loop over the subdomains in the overall algorithm and by benefiting from the characteristics of the Raviart-Thomas finite element. The new parallel algorithm still allows us to locally adapt the numerical scheme (mesh, finite element order). However, it can be significantly optimized for the matching grid case. The good behavior of the new parallelization scheme is demonstrated for the matching grid case on several hundreds of nodes for computations based on a pin-by-pin discretization.

Comparison of two Galerkin quadrature methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Morel, Jim E.; Warsa, James; Franke, Brian C.

Here, we compare two methods for generating Galerkin quadratures. In method 1, the standard S N method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard S N method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwisemore » sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding S N equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.« less

CCOMP: An efficient algorithm for complex roots computation of determinantal equations

NASA Astrophysics Data System (ADS)

Zouros, Grigorios P.

2018-01-01

In this paper a free Python algorithm, entitled CCOMP (Complex roots COMPutation), is developed for the efficient computation of complex roots of determinantal equations inside a prescribed complex domain. The key to the method presented is the efficient determination of the candidate points inside the domain which, in their close neighborhood, a complex root may lie. Once these points are detected, the algorithm proceeds to a two-dimensional minimization problem with respect to the minimum modulus eigenvalue of the system matrix. In the core of CCOMP exist three sub-algorithms whose tasks are the efficient estimation of the minimum modulus eigenvalues of the system matrix inside the prescribed domain, the efficient computation of candidate points which guarantee the existence of minima, and finally, the computation of minima via bound constrained minimization algorithms. Theoretical results and heuristics support the development and the performance of the algorithm, which is discussed in detail. CCOMP supports general complex matrices, and its efficiency, applicability and validity is demonstrated to a variety of microwave applications.

New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data

NASA Astrophysics Data System (ADS)

Audibert, Lorenzo; Cakoni, Fioralba; Haddar, Houssem

2017-12-01

In this paper we develop a general mathematical framework to determine interior eigenvalues from a knowledge of the modified far field operator associated with an unknown (anisotropic) inhomogeneity. The modified far field operator is obtained by subtracting from the measured far field operator the computed far field operator corresponding to a well-posed scattering problem depending on one (possibly complex) parameter. Injectivity of this modified far field operator is related to an appropriate eigenvalue problem whose eigenvalues can be determined from the scattering data, and thus can be used to obtain information about material properties of the unknown inhomogeneity. We discuss here two examples of such modification leading to a Steklov eigenvalue problem, and a new type of the transmission eigenvalue problem. We present some numerical examples demonstrating the viability of our method for determining the interior eigenvalues form far field data.

Eigenvalues of the Wentzell-Laplace operator and of the fourth order Steklov problems

NASA Astrophysics Data System (ADS)

Xia, Changyu; Wang, Qiaoling

2018-05-01

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.

Studying relaxation phenomena via effective master equations

NASA Astrophysics Data System (ADS)

Chan, David; Wan, Jones T. K.; Chu, L. L.; Yu, K. W.

2000-04-01

The real-time dynamics of various relaxation phenomena can be conveniently formulated by a master equation with the enumeration of transition rates between given classes of conformations. To study the relaxation time towards equilibrium, it suffices to solve for the second largest eigenvalue of the resulting eigenvalue equation. Generally speaking, there is no analytic solution for the dynamic equation. Mean-field approaches generally yield misleading results while the presumably exact Monte-Carlo methods require prohibitive time steps in most real systems. In this work, we propose an exact decimation procedure for reducing the number of conformations significantly, while there is no loss of information, i.e., the reduced (or effective) equation is an exact transformed version of the original one. However, we have to pay the price: the initial Markovianity of the evolution equation is lost and the reduced equation contains memory terms in the transition rates. Since the transformed equation has significantly reduced number of degrees of freedom, the systems can readily be diagonalized by iterative means, to obtain the exact second largest eigenvalue and hence the relaxation time. The decimation method has been applied to various relaxation equations with generally desirable results. The advantages and limitations of the method will be discussed.

Computation of free oscillations of the earth

USGS Publications Warehouse

Buland, Raymond P.; Gilbert, F.

1984-01-01

Although free oscillations of the Earth may be computed by many different methods, numerous practical considerations have led us to use a Rayleigh-Ritz formulation with piecewise cubic Hermite spline basis functions. By treating the resulting banded matrix equation as a generalized algebraic eigenvalue problem, we are able to achieve great accuracy and generality and a high degree of automation at a reasonable cost. ?? 1984.

Bethe-Salpeter Eigenvalue Solver Package (BSEPACK) v0.1

DOE Office of Scientific and Technical Information (OSTI.GOV)

SHAO, MEIYEU; YANG, CHAO

2017-04-25

The BSEPACK contains a set of subroutines for solving the Bethe-Salpeter Eigenvalue (BSE) problem. This type of problem arises in this study of optical excitation of nanoscale materials. The BSE problem is a structured non-Hermitian eigenvalue problem. The BSEPACK software can be used to compute all or subset of eigenpairs of a BSE Hamiltonian. It can also be used to compute the optical absorption spectrum without computing BSE eigenvalues and eigenvectors explicitly. The package makes use of the ScaLAPACK, LAPACK and BLAS.

Self-Similar Compressible Free Vortices

NASA Technical Reports Server (NTRS)

vonEllenrieder, Karl

1998-01-01

Lie group methods are used to find both exact and numerical similarity solutions for compressible perturbations to all incompressible, two-dimensional, axisymmetric vortex reference flow. The reference flow vorticity satisfies an eigenvalue problem for which the solutions are a set of two-dimensional, self-similar, incompressible vortices. These solutions are augmented by deriving a conserved quantity for each eigenvalue, and identifying a Lie group which leaves the reference flow equations invariant. The partial differential equations governing the compressible perturbations to these reference flows are also invariant under the action of the same group. The similarity variables found with this group are used to determine the decay rates of the velocities and thermodynamic variables in the self-similar flows, and to reduce the governing partial differential equations to a set of ordinary differential equations. The ODE's are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are used to examine the dependencies of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Also, experimental data on compressible free vortex flow are compared to the analytical results, the evolution of vortices from initial states which are not self-similar is discussed, and the energy transfer in a slightly-compressible vortex is considered.

Eigenmodes of Ducted Flows With Radially-Dependent Axial and Swirl Velocity Components

NASA Technical Reports Server (NTRS)

Kousen, Kenneth A.

1999-01-01

This report characterizes the sets of small disturbances possible in cylindrical and annular ducts with mean flow whose axial and tangential components vary arbitrarily with radius. The linearized equations of motion are presented and discussed, and then exponential forms for the axial, circumferential, and time dependencies of any unsteady disturbances are assumed. The resultant equations form a generalized eigenvalue problem, the solution of which yields the axial wavenumbers and radial mode shapes of the unsteady disturbances. Two numerical discretizations are applied to the system of equations: (1) a spectral collocation technique based on Chebyshev polynomial expansions on the Gauss-Lobatto points, and (2) second and fourth order finite differences on uniform grids. The discretized equations are solved using a standard eigensystem package employing the QR algorithm. The eigenvalues fall into two primary categories: a discrete set (analogous to the acoustic modes found in uniform mean flows) and a continuous band (analogous to convected disturbances in uniform mean flows) where the phase velocities of the disturbances correspond to the local mean flow velocities. Sample mode shapes and eigensystem distributions are presented for both sheared axial and swirling flows. The physics of swirling flows is examined with reference to hydrodynamic stability and completeness of the eigensystem expansions. The effect of assuming exponential dependence in the axial direction is discussed.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bai, Zhaojun; Yang, Chao

What is common among electronic structure calculation, design of MEMS devices, vibrational analysis of high speed railways, and simulation of the electromagnetic field of a particle accelerator? The answer: they all require solving large scale nonlinear eigenvalue problems. In fact, these are just a handful of examples in which solving nonlinear eigenvalue problems accurately and efficiently is becoming increasingly important. Recognizing the importance of this class of problems, an invited minisymposium dedicated to nonlinear eigenvalue problems was held at the 2005 SIAM Annual Meeting. The purpose of the minisymposium was to bring together numerical analysts and application scientists to showcasemore » some of the cutting edge results from both communities and to discuss the challenges they are still facing. The minisymposium consisted of eight talks divided into two sessions. The first three talks focused on a type of nonlinear eigenvalue problem arising from electronic structure calculations. In this type of problem, the matrix Hamiltonian H depends, in a non-trivial way, on the set of eigenvectors X to be computed. The invariant subspace spanned by these eigenvectors also minimizes a total energy function that is highly nonlinear with respect to X on a manifold defined by a set of orthonormality constraints. In other applications, the nonlinearity of the matrix eigenvalue problem is restricted to the dependency of the matrix on the eigenvalues to be computed. These problems are often called polynomial or rational eigenvalue problems In the second session, Christian Mehl from Technical University of Berlin described numerical techniques for solving a special type of polynomial eigenvalue problem arising from vibration analysis of rail tracks excited by high-speed trains.« less

A numerical projection technique for large-scale eigenvalue problems

NASA Astrophysics Data System (ADS)

Gamillscheg, Ralf; Haase, Gundolf; von der Linden, Wolfgang

2011-10-01

We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity is constructed by projecting out high energy degrees of freedom and in turn solving the resulting model by some standard eigenvalue solver. Here we introduce a generalization of this idea, where both steps are performed numerically and which in contrast to the standard projection technique converges in principle to the exact eigenvalues. This approach is not just applicable to eigenvalue problems encountered in many-body systems but also in other areas of research that result in large-scale eigenvalue problems for matrices which have, roughly speaking, mostly a pronounced dominant diagonal part. We will present detailed studies of the approach guided by two many-body models.

Coupled radial Schrödinger equations written as Dirac-type equations: application to an amplitude-phase approach

NASA Astrophysics Data System (ADS)

Thylwe, Karl-Erik; McCabe, Patrick

2012-04-01

The classical amplitude-phase method due to Milne, Wilson, Young and Wheeler in the 1930s is known to be a powerful computational tool for determining phase shifts and energy eigenvalues in cases where a sufficiently slowly varying amplitude function can be found. The key for the efficient computations is that the original single-state radial Schrödinger equation is transformed to a nonlinear equation, the Milne equation. Such an equation has solutions that may or may not oscillate, depending on boundary conditions, which then requires a robust recipe for locating the (optimal) ‘almost constant’ solutions for its use in the method. For scattering problems the solutions of the amplitude equations always approach constants as the radial distance r tends to infinity, and there is no problem locating the ‘optimal’ amplitude functions from a low-order semiclassical approximation. In the present work, the amplitude-phase approach is generalized to two coupled Schrödinger equations similar to an earlier generalization to radial Dirac equations. The original scalar amplitude then becomes a vector quantity, and the original Milne equation is generalized accordingly. Numerical applications to resonant electron-atom scattering are illustrated.

Compressed modes for variational problems in mathematics and physics

PubMed Central

Ozoliņš, Vidvuds; Lai, Rongjie; Caflisch, Russel; Osher, Stanley

2013-01-01

This article describes a general formalism for obtaining spatially localized (“sparse”) solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger’s equation in quantum mechanics. Sparsity is achieved by adding an regularization term to the variational principle, which is shown to yield solutions with compact support (“compressed modes”). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size. PMID:24170861

Compressed modes for variational problems in mathematics and physics.

PubMed

Ozolins, Vidvuds; Lai, Rongjie; Caflisch, Russel; Osher, Stanley

2013-11-12

This article describes a general formalism for obtaining spatially localized ("sparse") solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger's equation in quantum mechanics. Sparsity is achieved by adding an regularization term to the variational principle, which is shown to yield solutions with compact support ("compressed modes"). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size.

Analysis of the Hessian for Aerodynamic Optimization: Inviscid Flow

NASA Technical Reports Server (NTRS)

Arian, Eyal; Ta'asan, Shlomo

1996-01-01

In this paper we analyze inviscid aerodynamic shape optimization problems governed by the full potential and the Euler equations in two and three dimensions. The analysis indicates that minimization of pressure dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and the Euler equations. However the optimization problems in two and three dimensions are inherently different. While the two dimensional optimization problems are well-posed the three dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. A natural choice of such a regularization is derived. The analysis also gives an estimate of the Hessian's condition number which implies that the problems at hand are ill-conditioned. Infinite dimensional approximations for the Hessians are constructed and preconditioners for gradient based methods are derived from these approximate Hessians.

A new mathematical adjoint for the modified SAAF -SN equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schunert, Sebastian; Wang, Yaqi; Martineau, Richard

2015-01-01

We present a new adjoint FEM weak form, which can be directly used for evaluating the mathematical adjoint, suitable for perturbation calculations, of the self-adjoint angular flux SN equations (SAAF -SN) without construction and transposition of the underlying coefficient matrix. Stabilization schemes incorporated in the described SAAF -SN method make the mathematical adjoint distinct from the physical adjoint, i.e. the solution of the continuous adjoint equation with SAAF -SN . This weak form is implemented into RattleSnake, the MOOSE (Multiphysics Object-Oriented Simulation Environment) based transport solver. Numerical results verify the correctness of the implementation and show its utility both formore » fixed source and eigenvalue problems.« less

On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains

NASA Astrophysics Data System (ADS)

Cantrell, Robert Stephen; Cosner, Chris

We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.

Stability analysis of ultrasound thick-shell contrast agents

PubMed Central

Lu, Xiaozhen; Chahine, Georges L.; Hsiao, Chao-Tsung

2012-01-01

The stability of thick shell encapsulated bubbles is studied analytically. 3-D small perturbations are introduced to the spherical oscillations of a contrast agent bubble in response to a sinusoidal acoustic field with different amplitudes of excitation. The equations of the perturbation amplitudes are derived using asymptotic expansions and linear stability analysis is then applied to the resulting differential equations. The stability of the encapsulated microbubbles to nonspherical small perturbations is examined by solving an eigenvalue problem. The approach then identifies the fastest growing perturbations which could lead to the breakup of the encapsulated microbubble or contrast agent. PMID:22280568

Viscoelastic Damping of Turbine and Compressor Blade Vibrations.

DTIC Science & Technology

1982-03-01

and Materials Conference, Atlanta, GA: 6 April 1981. 6. Balfour, A. and D.H. Marwick. Pro-ramming in Standard FORTRAN 77. New York: North- Holand Inc...EIGENVALUE PROBLEM RESULTING FROM THE EQUATION OF C MOTION IN THE ANALYTICAL SECTION REAL Gl,G2,C,T,L,M,K,J, F1 ,F2,F3,F4,X1,X2 COMPLEX Z,Z1,B,D,F,W COMMON

A study of the control problem of the shoot side environment delivery system of a closed crop growth research chamber

NASA Technical Reports Server (NTRS)

Blackwell, C. C.; Blackwell, A. L.

1992-01-01

The details of our initial study of the control problem of the crop shoot environment of a hypothetical closed crop growth research chamber (CGRC) are presented in this report. The configuration of the CGRC is hypothetical because neither a physical subject nor a design existed at the time the study began, a circumstance which is typical of large scale systems control studies. The basis of the control study is a mathematical model which was judged to adequately mimic the relevant dynamics of the system components considered necessary to provide acceptable realism in the representation. Control of pressure, temperature, and flow rate of the crop shoot environment, along with its oxygen, carbon dioxide, and water concentration is addressed. To account for mass exchange, the group of plants is represented in the model by a source of oxygen, a source of water vapor, and a sink for carbon dioxide. In terms of the thermal energy exchange, the group of plants is represented by a surface with an appropriate temperature. Most of the primitive equations about an experimental operating condition and a state variable representation which was extracted from the linearized equations are presented. Next, we present the results of a real Jordan decomposition and the repositioning of an undesirable eigenvalue via full state feedback. The state variable representation of the modeling system is of the nineteenth order and reflects the eleven control variables and eight system disturbances. Five real eigenvalues are very near zero, with one at zero, three having small magnitude positive values, and one having a small magnitude negative value. A Singular Value Decomposition analysis indicates that these non-zero eigenvalues are not results of numerical error.

Stability investigations of airfoil flow by global analysis

NASA Technical Reports Server (NTRS)

Morzynski, Marek; Thiele, Frank

1992-01-01

As the result of global, non-parallel flow stability analysis the single value of the disturbance growth-rate and respective frequency is obtained. This complex value characterizes the stability of the whole flow configuration and is not referred to any particular flow pattern. The global analysis assures that all the flow elements (wake, boundary and shear layer) are taken into account. The physical phenomena connected with the wake instability are properly reproduced by the global analysis. This enhances the investigations of instability of any 2-D flows, including ones in which the boundary layer instability effects are known to be of dominating importance. Assuming fully 2-D disturbance form, the global linear stability problem is formulated. The system of partial differential equations is solved for the eigenvalues and eigenvectors. The equations, written in the pure stream function formulation, are discretized via FDM using a curvilinear coordinate system. The complex eigenvalues and corresponding eigenvectors are evaluated by an iterative method. The investigations performed for various Reynolds numbers emphasize that the wake instability develops into the Karman vortex street. This phenomenon is shown to be connected with the first mode obtained from the non-parallel flow stability analysis. The higher modes are reflecting different physical phenomena as for example Tollmien-Schlichting waves, originating in the boundary layer and having the tendency to emerge as instabilities for the growing Reynolds number. The investigations are carried out for a circular cylinder, oblong ellipsis and airfoil. It is shown that the onset of the wake instability, the waves in the boundary layer, the shear layer instability are different solutions of the same eigenvalue problem, formulated using the non-parallel theory. The analysis offers large potential possibilities as the generalization of methods used till now for the stability analysis.

Gravitational lensing by eigenvalue distributions of random matrix models

NASA Astrophysics Data System (ADS)

Martínez Alonso, Luis; Medina, Elena

2018-05-01

We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles.

A Novel Hyperbolization Procedure for The Two-Phase Six-Equation Flow Model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Samet Y. Kadioglu; Robert Nourgaliev; Nam Dinh

2011-10-01

We introduce a novel approach for the hyperbolization of the well-known two-phase six equation flow model. The six-equation model has been frequently used in many two-phase flow applications such as bubbly fluid flows in nuclear reactors. One major drawback of this model is that it can be arbitrarily non-hyperbolic resulting in difficulties such as numerical instability issues. Non-hyperbolic behavior can be associated with complex eigenvalues that correspond to characteristic matrix of the system. Complex eigenvalues are often due to certain flow parameter choices such as the definition of inter-facial pressure terms. In our method, we prevent the characteristic matrix receivingmore » complex eigenvalues by fine tuning the inter-facial pressure terms with an iterative procedure. In this way, the characteristic matrix possesses all real eigenvalues meaning that the characteristic wave speeds are all real therefore the overall two-phase flowmodel becomes hyperbolic. The main advantage of this is that one can apply less diffusive highly accurate high resolution numerical schemes that often rely on explicit calculations of real eigenvalues. We note that existing non-hyperbolic models are discretized mainly based on low order highly dissipative numerical techniques in order to avoid stability issues.« less

Optical reflection from planetary surfaces as an operator-eigenvalue problem

USGS Publications Warehouse

Wildey, R.L.

1986-01-01

The understanding of quantum mechanical phenomena has come to rely heavily on theory framed in terms of operators and their eigenvalue equations. This paper investigates the utility of that technique as related to the reciprocity principle in diffuse reflection. The reciprocity operator is shown to be unitary and Hermitian; hence, its eigenvectors form a complete orthonormal basis. The relevant eigenvalue is found to be infinitely degenerate. A superposition of the eigenfunctions found from solution by separation of variables is inadequate to form a general solution that can be fitted to a one-dimensional boundary condition, because the difficulty of resolving the reciprocity operator into a superposition of independent one-dimensional operators has yet to be overcome. A particular lunar application in the form of a failed prediction of limb-darkening of the full Moon from brightness versus phase illustrates this problem. A general solution is derived which fully exploits the determinative powers of the reciprocity operator as an unresolved two-dimensional operator. However, a solution based on a sum of one-dimensional operators, if possible, would be much more powerful. A close association is found between the reciprocity operator and the particle-exchange operator of quantum mechanics, which may indicate the direction for further successful exploitation of the approach based on the operational calculus. ?? 1986 D. Reidel Publishing Company.

Optimal control of coupled parabolic-hyperbolic non-autonomous PDEs: infinite-dimensional state-space approach

NASA Astrophysics Data System (ADS)

Aksikas, I.; Moghadam, A. Alizadeh; Forbes, J. F.

2018-04-01

This paper deals with the design of an optimal state-feedback linear-quadratic (LQ) controller for a system of coupled parabolic-hypebolic non-autonomous partial differential equations (PDEs). The infinite-dimensional state space representation and the corresponding operator Riccati differential equation are used to solve the control problem. Dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the LQ-optimal control problem and also to guarantee the exponential stability of the closed-loop system. Thanks to the eigenvalues and eigenfunctions of the parabolic operator and also the fact that the hyperbolic-associated operator Riccati differential equation can be converted to a scalar Riccati PDE, an algorithm to solve the LQ control problem has been presented. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ optimal controller designed in the early portion of the paper is implemented for the original non-linear model. Numerical simulations are performed to show the controller performances.

Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering

DOE PAGES

Willert, Jeffrey; Park, H.; Taitano, William

2015-11-01

High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Lastly, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cai, Yunfeng, E-mail: yfcai@math.pku.edu.cn; Department of Computer Science, University of California, Davis 95616; Bai, Zhaojun, E-mail: bai@cs.ucdavis.edu

2013-12-15

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for ab initio electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal blockmore » preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods.« less

A divide and conquer approach to the nonsymmetric eigenvalue problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1991-01-01

Serial computation combined with high communication costs on distributed-memory multiprocessors make parallel implementations of the QR method for the nonsymmetric eigenvalue problem inefficient. This paper introduces an alternative algorithm for the nonsymmetric tridiagonal eigenvalue problem based on rank two tearing and updating of the matrix. The parallelism of this divide and conquer approach stems from independent solution of the updating problems. 11 refs.

Neural network approach for the calculation of potential coefficients in quantum mechanics

NASA Astrophysics Data System (ADS)

Ossandón, Sebastián; Reyes, Camilo; Cumsille, Patricio; Reyes, Carlos M.

2017-05-01

A numerical method based on artificial neural networks is used to solve the inverse Schrödinger equation for a multi-parameter class of potentials. First, the finite element method was used to solve repeatedly the direct problem for different parametrizations of the chosen potential function. Then, using the attainable eigenvalues as a training set of the direct radial basis neural network a map of new eigenvalues was obtained. This relationship was later inverted and refined by training an inverse radial basis neural network, allowing the calculation of the unknown parameters and therefore estimating the potential function. Three numerical examples are presented in order to prove the effectiveness of the method. The results show that the method proposed has the advantage to use less computational resources without a significant accuracy loss.

Determination of gas & liquid two-phase flow regime transitions in wellbore annulus by virtual mass force coefficient when gas cut

NASA Astrophysics Data System (ADS)

Qu, Junbo; Yan, Tie; Sun, Xiaofeng; Chen, Ye; Pan, Yi

2017-10-01

With the development of drilling technology to deeper stratum, overflowing especially gas cut occurs frequently, and then flow regime in wellbore annulus is from the original drilling fluid single-phase flow into gas & liquid two-phase flow. By using averaged two-fluid model equations and the basic principle of fluid mechanics to establish the continuity equations and momentum conservation equations of gas phase & liquid phase respectively. Relationship between pressure and density of gas & liquid was introduced to obtain hyperbolic equation, and get the expression of the dimensionless eigenvalue of the equation by using the characteristic line method, and analyze wellbore flow regime to get the critical gas content under different virtual mass force coefficients. Results show that the range of equation eigenvalues is getting smaller and smaller with the increase of gas content. When gas content reaches the critical point, the dimensionless eigenvalue of equation has no real solution, and the wellbore flow regime changed from bubble flow to bomb flow. When virtual mass force coefficients are 0.50, 0.60, 0.70 and 0.80 respectively, the critical gas contents are 0.32, 0.34, 0.37 and 0.39 respectively. The higher the coefficient of virtual mass force, the higher gas content in wellbore corresponding to the critical point of transition flow regime, which is in good agreement with previous experimental results. Therefore, it is possible to determine whether there is a real solution of the dimensionless eigenvalue of equation by virtual mass force coefficient and wellbore gas content, from which we can obtain the critical condition of wellbore flow regime transformation. It can provide theoretical support for the accurate judgment of the annular flow regime.

Bifurcating fronts for the Taylor-Couette problem in infinite cylinders

NASA Astrophysics Data System (ADS)

Hărăguş-Courcelle, M.; Schneider, G.

We show the existence of bifurcating fronts for the weakly unstable Taylor-Couette problem in an infinite cylinder. These fronts connect a stationary bifurcating pattern, here the Taylor vortices, with the trivial ground state, here the Couette flow. In order to show the existence result we improve a method which was already used in establishing the existence of bifurcating fronts for the Swift-Hohenberg equation by Collet and Eckmann, 1986, and by Eckmann and Wayne, 1991. The existence proof is based on spatial dynamics and center manifold theory. One of the difficulties in applying center manifold theory comes from an infinite number of eigenvalues on the imaginary axis for vanishing bifurcation parameter. But nevertheless, a finite dimensional reduction is possible, since the eigenvalues leave the imaginary axis with different velocities, if the bifurcation parameter is increased. In contrast to previous work we have to use normalform methods and a non-standard cut-off function to obtain a center manifold which is large enough to contain the bifurcating fronts.

Learning SVM in Kreĭn Spaces.

PubMed

Loosli, Gaelle; Canu, Stephane; Ong, Cheng Soon

2016-06-01

This paper presents a theoretical foundation for an SVM solver in Kreĭn spaces. Up to now, all methods are based either on the matrix correction, or on non-convex minimization, or on feature-space embedding. Here we justify and evaluate a solution that uses the original (indefinite) similarity measure, in the original Kreĭn space. This solution is the result of a stabilization procedure. We establish the correspondence between the stabilization problem (which has to be solved) and a classical SVM based on minimization (which is easy to solve). We provide simple equations to go from one to the other (in both directions). This link between stabilization and minimization problems is the key to obtain a solution in the original Kreĭn space. Using KSVM, one can solve SVM with usually troublesome kernels (large negative eigenvalues or large numbers of negative eigenvalues). We show experiments showing that our algorithm KSVM outperforms all previously proposed approaches to deal with indefinite matrices in SVM-like kernel methods.

Diffusion with Varying Drag; the Runaway Problem.

NASA Astrophysics Data System (ADS)

Rollins, David Kenneth

We study the motion of electrons in an ionized plasma of electrons and ions in an external electric field. A probability distribution function describes the electron motion and is a solution of a Fokker-Planck equation. In zero field, the solution approaches an equilibrium Maxwellian. For arbitrarily small field, electrons overcome the diffusive effects and are freely accelerated by the field. This is the electron runaway phenomenon. We treat the electric field as a small perturbation. We consider various diffusion coefficients for the one dimensional problem and determine the runaway current as a function of the field strength. Diffusion coefficients, non-zero on a finite interval are examined. Some non-trivial cases of these can be solved exactly in terms of known special functions. The more realistic case where the diffusion coefficient decays with velocity are then considered. To determine the runaway current, the equivalent Schrodinger eigenvalue problem is analysed. The smallest eigenvalue is shown to be equal to the runaway current. Using asymptotic matching a solution can be constructed which is then used to evaluate the runaway current. The runaway current is exponentially small as a function of field strength. This method is used to extract results from the three dimensional problem.

Computation of steady and unsteady quasi-one-dimensional viscous/inviscid interacting internal flows at subsonic, transonic, and supersonic Mach numbers

NASA Technical Reports Server (NTRS)

Swafford, Timothy W.; Huddleston, David H.; Busby, Judy A.; Chesser, B. Lawrence

1992-01-01

Computations of viscous-inviscid interacting internal flowfields are presented for steady and unsteady quasi-one-dimensional (Q1D) test cases. The unsteady Q1D Euler equations are coupled with integral boundary-layer equations for unsteady, two-dimensional (planar or axisymmetric), turbulent flow over impermeable, adiabatic walls. The coupling methodology differs from that used in most techniques reported previously in that the above mentioned equation sets are written as a complete system and solved simultaneously; that is, the coupling is carried out directly through the equations as opposed to coupling the solutions of the different equation sets. Solutions to the coupled system of equations are obtained using both explicit and implicit numerical schemes for steady subsonic, steady transonic, and both steady and unsteady supersonic internal flowfields. Computed solutions are compared with measurements as well as Navier-Stokes and inverse boundary-layer methods. An analysis of the eigenvalues of the coefficient matrix associated with the quasi-linear form of the coupled system of equations indicates the presence of complex eigenvalues for certain flow conditions. It is concluded that although reasonable solutions can be obtained numerically, these complex eigenvalues contribute to the overall difficulty in obtaining numerical solutions to the coupled system of equations.

Gauge equivalence of the Gross Pitaevskii equation and the equivalent Heisenberg spin chain

NASA Astrophysics Data System (ADS)

Radha, R.; Kumar, V. Ramesh

2007-11-01

In this paper, we construct an equivalent spin chain for the Gross-Pitaevskii equation with quadratic potential and exponentially varying scattering lengths using gauge equivalence. We have then generated the soliton solutions for the spin components S3 and S-. We find that the spin solitons for S3 and S- can be compressed for exponentially growing eigenvalues while they broaden out for decaying eigenvalues.

Sum rules and other properties involving resonance projection operators. [for optical potential description of electron scattering from atoms and ions

NASA Technical Reports Server (NTRS)

Berk, A.; Temkin, A.

1985-01-01

A sum rule is derived for the auxiliary eigenvalues of an equation whose eigenspectrum pertains to projection operators which describe electron scattering from multielectron atoms and ions. The sum rule's right-hand side depends on an integral involving the target system eigenfunctions. The sum rule is checked for several approximations of the two-electron target. It is shown that target functions which have a unit eigenvalue in their auxiliary eigenspectrum do not give rise to well-defined projection operators except through a limiting process. For Hylleraas target approximations, the auxiliary equations are shown to contain an infinite spectrum. However, using a Rayleigh-Ritz variational principle, it is shown that a comparatively simple aproximation can exhaust the sum rule to better than five significant figures. The auxiliary Hylleraas equation is greatly simplified by conversion to a square root equation containing the same eigenfunction spectrum and from which the required eigenvalues are trivially recovered by squaring.

The problem of complex eigensystems in the semianalytical solution for advancement of time in solute transport simulations: a new method using real arithmetic

USGS Publications Warehouse

Umari, Amjad M.J.; Gorelick, Steven M.

1986-01-01

In the numerical modeling of groundwater solute transport, explicit solutions may be obtained for the concentration field at any future time without computing concentrations at intermediate times. The spatial variables are discretized and time is left continuous in the governing differential equation. These semianalytical solutions have been presented in the literature and involve the eigensystem of a coefficient matrix. This eigensystem may be complex (i.e., have imaginary components) due to the asymmetry created by the advection term in the governing advection-dispersion equation. Previous investigators have either used complex arithmetic to represent a complex eigensystem or chosen large dispersivity values for which the imaginary components of the complex eigenvalues may be ignored without significant error. It is shown here that the error due to ignoring the imaginary components of complex eigenvalues is large for small dispersivity values. A new algorithm that represents the complex eigensystem by converting it to a real eigensystem is presented. The method requires only real arithmetic.

Computing the Evans function via solving a linear boundary value ODE

NASA Astrophysics Data System (ADS)

Wahl, Colin; Nguyen, Rose; Ventura, Nathaniel; Barker, Blake; Sandstede, Bjorn

2015-11-01

Determining the stability of traveling wave solutions to partial differential equations can oftentimes be computationally intensive but of great importance to understanding the effects of perturbations on the physical systems (chemical reactions, hydrodynamics, etc.) they model. For waves in one spatial dimension, one may linearize around the wave and form an Evans function - an analytic Wronskian-like function which has zeros that correspond in multiplicity to the eigenvalues of the linearized system. If eigenvalues with a positive real part do not exist, the traveling wave will be stable. Two methods exist for calculating the Evans function numerically: the exterior-product method and the method of continuous orthogonalization. The first is numerically expensive, and the second reformulates the originally linear system as a nonlinear system. We develop a new algorithm for computing the Evans function through appropriate linear boundary-value problems. This algorithm is cheaper than the previous methods, and we prove that it preserves analyticity of the Evans function. We also provide error estimates and implement it on some classical one- and two-dimensional systems, one being the Swift-Hohenberg equation in a channel, to show the advantages.

Reactor Application for Coaching Newbies

DOE Office of Scientific and Technical Information (OSTI.GOV)

2015-06-17

RACCOON is a Moose based reactor physics application designed to engage undergraduate and first-year graduate students. The code contains capabilities to solve the multi group Neutron Diffusion equation in eigenvalue and fixed source form and will soon have a provision to provide simple thermal feedback. These capabilities are sufficient to solve example problems found in Duderstadt & Hamilton (the typical textbook of senior level reactor physics classes). RACCOON does not contain any advanced capabilities as found in YAK.

Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements

NASA Astrophysics Data System (ADS)

Wu, Sheng-Jhih; Chu, Moody T.

2017-08-01

An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.

A High Frequency Model of Cascade Noise

NASA Technical Reports Server (NTRS)

Envia, Edmane

1998-01-01

Closed form asymptotic expressions for computing high frequency noise generated by an annular cascade in an infinite duct containing a uniform flow are presented. There are two new elements in this work. First, the annular duct mode representation does not rely on the often-used Bessel function expansion resulting in simpler expressions for both the radial eigenvalues and eigenfunctions of the duct. In particular, the new representation provides an explicit approximate formula for the radial eigenvalues obviating the need for solutions of the transcendental annular duct eigenvalue equation. Also, the radial eigenfunctions are represented in terms of exponentials eliminating the numerical problems associated with generating the Bessel functions on a computer. The second new element is the construction of an unsteady response model for an annular cascade. The new construction satisfies the boundary conditions on both the cascade and duct walls simultaneously adding a new level of realism to the noise calculations. Preliminary results which demonstrate the effectiveness of the new elements are presented. A discussion of the utility of the asymptotic formulas for calculating cascade discrete tone as well as broadband noise is also included.

Comment on “Approximate solutions of the Dirac equation for the Rosen-Morse potential including the spin-orbit centrifugal term” [J. Math. Phys. 51, 023525 (2010)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ghoumaid, A.; Benamira, F.; Guechi, L.

2016-02-15

It is shown that the application of the Nikiforov-Uvarov method by Ikhdair for solving the Dirac equation with the radial Rosen-Morse potential plus the spin-orbit centrifugal term is inadequate because the required conditions are not satisfied. The energy spectra given is incorrect and the wave functions are not physically acceptable. We clarify the problem and prove that the spinor wave functions are expressed in terms of the generalized hypergeometric functions {sub 2}F{sub 1}(a, b, c; z). The energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric function.

Mixed finite-difference scheme for free vibration analysis of noncircular cylinders

NASA Technical Reports Server (NTRS)

Noor, A. K.; Stephens, W. B.

1973-01-01

A mixed finite-difference scheme is presented for the free-vibration analysis of simply supported closed noncircular cylindrical shells. The problem is formulated in terms of eight first-order differential equations in the circumferential coordinate which possess a symmetric coefficient matrix and are free of the derivatives of the elastic and geometric characteristics of the shell. In the finite-difference discretization, two interlacing grids are used for the different fundamental unknowns in such a way as to avoid averaging in the difference-quotient expressions used for the first derivative. The resulting finite-difference equations are symmetric. The inverse-power method is used for obtaining the eigenvalues and eigenvectors.

Stability analysis of ultrasound thick-shell contrast agents.

PubMed

Lu, Xiaozhen; Chahine, Georges L; Hsiao, Chao-Tsung

2012-01-01

The stability of thick shell encapsulated bubbles is studied analytically. 3-D small perturbations are introduced to the spherical oscillations of a contrast agent bubble in response to a sinusoidal acoustic field with different amplitudes of excitation. The equations of the perturbation amplitudes are derived using asymptotic expansions and linear stability analysis is then applied to the resulting differential equations. The stability of the encapsulated microbubbles to nonspherical small perturbations is examined by solving an eigenvalue problem. The approach then identifies the fastest growing perturbations which could lead to the breakup of the encapsulated microbubble or contrast agent. © 2012 Acoustical Society of America.

Nonlinear travelling waves in rotating Hagen–Poiseuille flow

NASA Astrophysics Data System (ADS)

Pier, Benoît; Govindarajan, Rama

2018-03-01

The dynamics of viscous flow through a rotating pipe is considered. Small-amplitude stability characteristics are obtained by linearizing the Navier–Stokes equations around the base flow and solving the resulting eigenvalue problems. For linearly unstable configurations, the dynamics leads to fully developed finite-amplitude perturbations that are computed by direct numerical simulations of the complete Navier–Stokes equations. By systematically investigating all linearly unstable combinations of streamwise wave number k and azimuthal mode number m, for streamwise Reynolds numbers {{Re}}z ≤slant 500 and rotational Reynolds numbers {{Re}}{{Ω }} ≤slant 500, the complete range of nonlinear travelling waves is obtained and the associated flow fields are characterized.

Issac, Jason Cherian ses in transonic flow

NASA Technical Reports Server (NTRS)

Issac, Jason Cherion; Kapania, Rakesh K.

1993-01-01

Flutter analysis of a two degree of freedom airfoil in compressible flow is performed using a state-space representation of the unsteady aerodynamic behavior. Indicial response functions are used to represent the normal force and moment response of the airfoil. The structural equations of motion of the airfoil with bending and torsional degrees of freedom are coupled to the unsteady air loads and the aeroelastic system so modelled is solved as an eigenvalue problem to determine the stability. The aeroelastic equations are also directly integrated with respect to time and the time-domain results compared with the results from the eigenanalysis. A good agreement is obtained. The derivatives of the flutter speed obtained from the eigenanalysis are calculated with respect to the mass and stiffness parameters by both analytical and finite-difference methods for various transonic Mach numbers. The experience gained from the two degree of freedom model is applied to study the sensitivity of the flutter response of a wing with respect to various shape parameters. The parameters being considered are as follows: (1) aspect ratio; (2) surface area of the wing; (3) taper ratio; and (4) sweep. The wing deflections are represented by Chebyshev polynomials. The compressible aerodynamic state-space model used for the airfoil section is extended to represent the unsteady aerodynamic forces on a generally laminated tapered skewed wing. The aeroelastic equations are solved as an eigenvalue problem to determine the flutter speed of the wing. The derivatives of the flutter speed with respect to the shape parameters are calculated by both analytical and finite difference methods.

Numerical implementation of complex orthogonalization, parallel transport on Stiefel bundles, and analyticity

NASA Astrophysics Data System (ADS)

Avitabile, Daniele; Bridges, Thomas J.

2010-06-01

Numerical integration of complex linear systems of ODEs depending analytically on an eigenvalue parameter are considered. Complex orthogonalization, which is required to stabilize the numerical integration, results in non-analytic systems. It is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non-analytic characteristic function. The orthonormal systems are constructed using the geometry of Stiefel bundles. Different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one-form on the Stiefel bundle. For the numerical integration, Gauss-Legendre Runge-Kutta algorithms are the principal choice for preserving orthogonality, and performance results are shown for a range of GLRK methods. The theory and methods are tested by application to example boundary value problems including the Orr-Sommerfeld equation in hydrodynamic stability.

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

NASA Astrophysics Data System (ADS)

Wang, Yan; Zhang, Yufeng; Zhang, Xiangzhi

2016-09-01

We first introduced a linear stationary equation with a quadratic operator in ∂x and ∂y, then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

Calculation of transmission probability by solving an eigenvalue problem

NASA Astrophysics Data System (ADS)

Bubin, Sergiy; Varga, Kálmán

2010-11-01

The electron transmission probability in nanodevices is calculated by solving an eigenvalue problem. The eigenvalues are the transmission probabilities and the number of nonzero eigenvalues is equal to the number of open quantum transmission eigenchannels. The number of open eigenchannels is typically a few dozen at most, thus the computational cost amounts to the calculation of a few outer eigenvalues of a complex Hermitian matrix (the transmission matrix). The method is implemented on a real space grid basis providing an alternative to localized atomic orbital based quantum transport calculations. Numerical examples are presented to illustrate the efficiency of the method.

IONIZATION EQUILIBRIUM TIMESCALES IN COLLISIONAL PLASMAS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smith, Randall K.; Hughes, John P., E-mail: rsmith@cfa.harvard.ed, E-mail: jph@physics.rutgers.ed

2010-07-20

Astrophysical shocks or bursts from a photoionizing source can disturb the typical collisional plasma found in galactic interstellar media or the intergalactic medium. The spectrum emitted by this plasma contains diagnostics that have been used to determine the time since the disturbing event, although this determination becomes uncertain as the elements in the plasma return to ionization equilibrium. A general solution for the equilibrium timescale for each element arises from the elegant eigenvector method of solution to the problem of a non-equilibrium plasma described by Masai and Hughes and Helfand. In general, the ionization evolution of an element Z inmore » a constant electron temperature plasma is given by a coupled set of Z + 1 first-order differential equations. However, they can be recast as Z uncoupled first-order differential equations using an eigenvector basis for the system. The solution is then Z separate exponential functions, with the time constants given by the eigenvalues of the rate matrix. The smallest of these eigenvalues gives the scale of the slowest return to equilibrium independent of the initial conditions, while conversely the largest eigenvalue is the scale of the fastest change in the ion population. These results hold for an ionizing plasma, a recombining plasma, or even a plasma with random initial conditions, and will allow users of these diagnostics to determine directly if their best-fit result significantly limits the timescale since a disturbance or is so close to equilibrium as to include an arbitrarily long time.« less

A Computer Program for the Computation of Running Gear Temperatures Using Green's Function

NASA Technical Reports Server (NTRS)

Koshigoe, S.; Murdock, J. W.; Akin, L. S.; Townsend, D. P.

1996-01-01

A new technique has been developed to study two dimensional heat transfer problems in gears. This technique consists of transforming the heat equation into a line integral equation with the use of Green's theorem. The equation is then expressed in terms of eigenfunctions that satisfy the Helmholtz equation, and their corresponding eigenvalues for an arbitrarily shaped region of interest. The eigenfunction are obtalned by solving an intergral equation. Once the eigenfunctions are found, the temperature is expanded in terms of the eigenfunctions with unknown time dependent coefficients that can be solved by using Runge Kutta methods. The time integration is extremely efficient. Therefore, any changes in the time dependent coefficients or source terms in the boundary conditions do not impose a great computational burden on the user. The method is demonstrated by applying it to a sample gear tooth. Temperature histories at representative surface locatons are given.

Applied Computational Electromagnetics Society Journal, volume 9, number 1, March 1994

NASA Astrophysics Data System (ADS)

1994-03-01

The partial contents of this document include the following: On the Use of Bivariate Spline Interpolation of Slot Data in the Design of Slotted Waveguide Arrays; A Technique for Determining Non-Integer Eigenvalues for Solutions of Ordinary Differential Equations; Antenna Modeling and Characterization of a VLF Airborne Dual Trailing Wire Antenna System; Electromagnetic Scattering from Two-Dimensional Composite Objects; and Use of a Stealth Boundary with Finite Difference Frequency Domain Simulations of Simple Antenna Problems.

Renormalization of the unitary evolution equation for coined quantum walks

NASA Astrophysics Data System (ADS)

Boettcher, Stefan; Li, Shanshan; Portugal, Renato

2017-03-01

We consider discrete-time evolution equations in which the stochastic operator of a classical random walk is replaced by a unitary operator. Such a problem has gained much attention as a framework for coined quantum walks that are essential for attaining the Grover limit for quantum search algorithms in physically realizable, low-dimensional geometries. In particular, we analyze the exact real-space renormalization group (RG) procedure recently introduced to study the scaling of quantum walks on fractal networks. While this procedure, when implemented numerically, was able to provide some deep insights into the relation between classical and quantum walks, its analytic basis has remained obscure. Our discussion here is laying the groundwork for a rigorous implementation of the RG for this important class of transport and algorithmic problems, although some instances remain unresolved. Specifically, we find that the RG fixed-point analysis of the classical walk, which typically focuses on the dominant Jacobian eigenvalue {λ1} , with walk dimension dw\\text{RW}={{log}2}{λ1} , needs to be extended to include the subdominant eigenvalue {λ2} , such that the dimension of the quantum walk obtains dw\\text{QW}={{log}2}\\sqrt{{λ1}{λ2}} . With that extension, we obtain analytically previously conjectured results for dw\\text{QW} of Grover walks on all but one of the fractal networks that have been considered.

Linear Temporal Stability Analysis of a Low-Density Round Gas Jet Injected into a High-Density Gas

NASA Technical Reports Server (NTRS)

Lawson, Anthony L.; Parthasarathy, Ramkumar N.

2002-01-01

It has been observed in previous experimental studies that round helium jets injected into air display a repetitive structure for a long distance, somewhat similar to the buoyancy-induced flickering observed in diffusion flames. In order to investigate the influence of gravity on the near-injector development of the flow, a linear temporal stability analysis of a round helium jet injected into air was performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The temporal growth rates and the phase velocity of the disturbances were obtained. The temporal growth rates of the disturbances increased as the Froude number was reduced (i.e. gravitational effects increased), indicating the destabilizing role played by gravity.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

NASA Astrophysics Data System (ADS)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; Govind, Niranjan; Yang, Chao

2017-12-01

We present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.

Note on the eigensolution of a homogeneous equation with semi-infinite domain

NASA Technical Reports Server (NTRS)

Wadia, A. R.

1980-01-01

The 'variation-iteration' method using Green's functions to find the eigenvalues and the corresponding eigenfunctions of a homogeneous Fredholm integral equation is employed for the stability analysis of fluid hydromechanics problems with a semiinfinite (infinite) domain of application. The objective of the study is to develop a suitable numerical approach to the solution of such equations in order to better understand the full set of equations for 'real-world' flow models. The study involves a search for a suitable value of the length of the domain which is a fair finite approximation to infinity, which makes the eigensolution an approximation dependent on the length of the interval chosen. In the examples investigated y = 1 = a seems to be the best approximation of infinity; for y greater than unity this method fails due to the polynomial nature of Green's functions.

Optimal linear-quadratic control of coupled parabolic-hyperbolic PDEs

NASA Astrophysics Data System (ADS)

Aksikas, I.; Moghadam, A. Alizadeh; Forbes, J. F.

2017-10-01

This paper focuses on the optimal control design for a system of coupled parabolic-hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear-quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ-optimal controller designed in the early portion of the paper is implemented for the original nonlinear model. Numerical simulations are performed to show the controller performances.

An accurate method for solving a class of fractional Sturm-Liouville eigenvalue problems

NASA Astrophysics Data System (ADS)

Kashkari, Bothayna S. H.; Syam, Muhammed I.

2018-06-01

This article is devoted to both theoretical and numerical study of the eigenvalues of nonsingular fractional second-order Sturm-Liouville problem. In this paper, we implement a fractional-order Legendre Tau method to approximate the eigenvalues. This method transforms the Sturm-Liouville problem to a sparse nonsingular linear system which is solved using the continuation method. Theoretical results for the considered problem are provided and proved. Numerical results are presented to show the efficiency of the proposed method.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-innermore » product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.« less

Finite-difference solution of the compressible stability eigenvalue problem

NASA Technical Reports Server (NTRS)

Malik, M. R.

1982-01-01

A compressible stability analysis computer code is developed. The 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 a Laminar Flow Control (LFC) swept wing.

The nonconforming virtual element method for eigenvalue problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gardini, Francesca; Manzini, Gianmarco; Vacca, Giuseppe

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L 2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problems. The proposed schemes provide a correct approximation of the spectrum and we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numericalmore » tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.« less

Solving complex band structure problems with the FEAST eigenvalue algorithm

NASA Astrophysics Data System (ADS)

Laux, S. E.

2012-08-01

With straightforward extension, the FEAST eigenvalue algorithm [Polizzi, Phys. Rev. B 79, 115112 (2009)] is capable of solving the generalized eigenvalue problems representing traveling-wave problems—as exemplified by the complex band-structure problem—even though the matrices involved are complex, non-Hermitian, and singular, and hence outside the originally stated range of applicability of the algorithm. The obtained eigenvalues/eigenvectors, however, contain spurious solutions which must be detected and removed. The efficiency and parallel structure of the original algorithm are unaltered. The complex band structures of Si layers of varying thicknesses and InAs nanowires of varying radii are computed as test problems.

Study of modal coupling procedures for the shuttle: A matrix method for damping synthesis

NASA Technical Reports Server (NTRS)

Hasselman, T. K.

1972-01-01

The damping method was applied successfully to real structures as well as analytical models. It depends on the ability to determine an appropriate modal damping matrix for each substructure. In the past, modal damping matrices were assumed diagonal for lack of being able to determine the coupling terms which are significant in the general case of nonproportional damping. This problem was overcome by formulating the damped equations of motion as a linear perturbation of the undamped equations for light structural damping. Damped modes are defined as complex vectors derived from the complex frequency response vectors of each substructure and are obtained directly from sinusoidal vibration tests. The damped modes are used to compute first order approximations to the modal damping matrices. The perturbation approach avoids ever having to solve a complex eigenvalue problem.

Supersymmetric symplectic quantum mechanics

NASA Astrophysics Data System (ADS)

de Menezes, Miralvo B.; Fernandes, M. C. B.; Martins, Maria das Graças R.; Santana, A. E.; Vianna, J. D. M.

2018-02-01

Symplectic Quantum Mechanics SQM considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville-von Neumann equation. In this article we extend the methods of supersymmetric quantum mechanics SUSYQM to SQM. With the purpose of applications in quantum systems, the factorization method of the quantum mechanical formalism is then set within supersymmetric SQM. A hierarchy of simpler hamiltonians is generated leading to new computation tools for solving the eigenvalue problem in SQM. We illustrate the results by computing the states and spectra of the problem of a charged particle in a homogeneous magnetic field as well as the corresponding Wigner function.

Eigenvalue asymptotics for the damped wave equation on metric graphs

NASA Astrophysics Data System (ADS)

Freitas, Pedro; Lipovský, Jiří

2017-09-01

We consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues. In the case of equilateral graphs and standard coupling conditions we show that there is only a finite number of high-frequency abscissas, whose location is solely determined by the averages of the damping terms on each edge. We further describe some of the possible behaviour when the edge lengths are no longer necessarily equal but remain commensurate.

A new method for computation of eigenvector derivatives with distinct and repeated eigenvalues in structural dynamic analysis

NASA Astrophysics Data System (ADS)

Li, Zhengguang; Lai, Siu-Kai; Wu, Baisheng

2018-07-01

Determining eigenvector derivatives is a challenging task due to the singularity of the coefficient matrices of the governing equations, especially for those structural dynamic systems with repeated eigenvalues. An effective strategy is proposed to construct a non-singular coefficient matrix, which can be directly used to obtain the eigenvector derivatives with distinct and repeated eigenvalues. This approach also has an advantage that only requires eigenvalues and eigenvectors of interest, without solving the particular solutions of eigenvector derivatives. The Symmetric Quasi-Minimal Residual (SQMR) method is then adopted to solve the governing equations, only the existing factored (shifted) stiffness matrix from an iterative eigensolution such as the subspace iteration method or the Lanczos algorithm is utilized. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner. Three numerical examples are given to illustrate the accuracy and validity of the proposed algorithm. Highly accurate approximations to the eigenvector derivatives are obtained within a few iteration steps, making a significant reduction of the computational effort. This method can be incorporated into a coupled eigensolver/derivative software module. In particular, it is applicable for finite element models with large sparse matrices.

Lie-Santilli isoapproach to the unification of gravity and electromagnetism

DOE Office of Scientific and Technical Information (OSTI.GOV)

Animalu, A.O.E.

1996-06-01

The author reviews the problem of Einstein`s original proposal for the unification of gravity and electromagnetism in space-time differential geometry along the lines of the recent contributions by A.A. Logunov, R.M. Santilli, D.F. Lopez and others. The author presents a new method of unification based on the Lie-Santilli isotopic theory whereby the unified field tensor g = (g{sub {mu}{nu}}) is constructed from the symmetric Riemannian gravitational tensor, g = (g{mu}{nu}), and the antisymmetric electromagnetic field tensor F = (F{sub {mu}{nu}}) via an isotopic lifting g {yields} {cflx g} = Fg of the type of Lax pairing, where det F {ne}more » 0, the unified field {cflx g} satisfies Logunov-Santilli equations while g and F are treated as Lax pair. Because of Santilli`s isotopic equivalence between Minkowskian and Riemannian geometries, the author infers that in the Minkowskian limit F = f, g = {eta}, the metric {eta} satisfies Lax`s equation of motion {partial_derivative}{eta}/{partial_derivative}t = f{eta} {minus} {eta}f which insures the conservation of the eigenvalues of g. The invariance of the electromagnetic group of transformations (F) in Minkowski space is determined by the eigenvalue equations, det (F{sub {mu}{nu}}){minus}{lambda}{eta}{sub {mu}{nu}} = 0, from which the author deduces a Lie-isotopic {open_quotes}extended{close_quotes} relativity principle. A wave equation for a spin-2 particle in the unified field is derived, and the experimental consequences of the theory are discussed.« less

Analytical spectrum for a Hamiltonian of quantum dots with Rashba spin-orbit coupling

NASA Astrophysics Data System (ADS)

Dossa, Anselme F.; Avossevou, Gabriel Y. H.

2014-12-01

We determine the analytical solution for a Hamiltonian describing a confined charged particle in a quantum dot, including Rashba spin-orbit coupling and Zeeman splitting terms. The approach followed in this paper is straightforward and uses the symmetrization of the wave function's components. The eigenvalue problem for the Hamiltonian in Bargmann's Hilbert space reduces to a system of coupled first-order differential equations. Then we exploit the symmetry in the system to obtain uncoupled second-order differential equations, which are found to be the Whittaker-Ince limit of the confluent Heun equations. Analytical expressions as well as numerical results are obtained for the spectrum. One of the main features of such models, namely, the level splitting, is present through the spectrum obtained in this paper.

Renormalization group procedure for potential -g/r2

NASA Astrophysics Data System (ADS)

Dawid, S. M.; Gonsior, R.; Kwapisz, J.; Serafin, K.; Tobolski, M.; Głazek, S. D.

2018-02-01

Schrödinger equation with potential - g /r2 exhibits a limit cycle, described in the literature in a broad range of contexts using various regularizations of the singularity at r = 0. Instead, we use the renormalization group transformation based on Gaussian elimination, from the Hamiltonian eigenvalue problem, of high momentum modes above a finite, floating cutoff scale. The procedure identifies a richer structure than the one we found in the literature. Namely, it directly yields an equation that determines the renormalized Hamiltonians as functions of the floating cutoff: solutions to this equation exhibit, in addition to the limit-cycle, also the asymptotic-freedom, triviality, and fixed-point behaviors, the latter in vicinity of infinitely many separate pairs of fixed points in different partial waves for different values of g.

Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity

NASA Astrophysics Data System (ADS)

Gómez, D.; Nazarov, S. A.; Pérez, M. E.

2018-04-01

We consider a homogenization Winkler-Steklov spectral problem that consists of the elasticity equations for a three-dimensional homogeneous anisotropic elastic body which has a plane part of the surface subject to alternating boundary conditions on small regions periodically placed along the plane. These conditions are of the Dirichlet type and of the Winkler-Steklov type, the latter containing the spectral parameter. The rest of the boundary of the body is fixed, and the period and size of the regions, where the spectral parameter arises, are of order ɛ . For fixed ɛ , the problem has a discrete spectrum, and we address the asymptotic behavior of the eigenvalues {β _k^ɛ }_{k=1}^{∞} as ɛ → 0. We show that β _k^ɛ =O(ɛ ^{-1}) for each fixed k, and we observe a common limit point for all the rescaled eigenvalues ɛ β _k^ɛ while we make it evident that, although the periodicity of the structure only affects the boundary conditions, a band-gap structure of the spectrum is inherited asymptotically. Also, we provide the asymptotic behavior for certain "groups" of eigenmodes.

The cosmological constant as an eigenvalue of a Sturm-Liouville problem

NASA Astrophysics Data System (ADS)

Astashenok, Artyom V.; Elizalde, Emilio; Yurov, Artyom V.

2014-01-01

It is observed that one of Einstein-Friedmann's equations has formally the aspect of a Sturm-Liouville problem, and that the cosmological constant, Λ, plays thereby the role of spectral parameter (what hints to its connection with the Casimir effect). The subsequent formulation of appropriate boundary conditions leads to a set of admissible values for Λ, considered as eigenvalues of the corresponding linear operator. Simplest boundary conditions are assumed, namely that the eigenfunctions belong to L 2 space, with the result that, when all energy conditions are satisfied, they yield a discrete spectrum for Λ>0 and a continuous one for Λ<0. A very interesting situation is seen to occur when the discrete spectrum contains only one point: then, there is the possibility to obtain appropriate cosmological conditions without invoking the anthropic principle. This possibility is shown to be realized in cyclic cosmological models, provided the potential of the matter field is similar to the potential of the scalar field. The dynamics of the universe in this case contains a sudden future singularity.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

In this article, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE PAGES

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; ...

2017-12-01

In this article, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE PAGES

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; ...

2017-08-24

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Analytic Regularity and Polynomial Approximation of Parametric and Stochastic Elliptic PDEs

DTIC Science & Technology

2010-05-31

Todor : Finite elements for elliptic problems with stochastic coefficients Comp. Meth. Appl. Mech. Engg. 194 (2005) 205-228. [14] R. Ghanem and P. Spanos...for elliptic partial differential equations with random input data SIAM J. Num. Anal. 46(2008), 2411–2442. [20] R. Todor , Robust eigenvalue computation...for smoothing operators, SIAM J. Num. Anal. 44(2006), 865– 878. [21] Ch. Schwab and R.A. Todor , Karhúnen-Loève Approximation of Random Fields by

Confining potential in momentum space

NASA Technical Reports Server (NTRS)

Norbury, John W.; Kahana, David E.; Maung, Khin Maung

1992-01-01

A method is presented for the solution in momentum space of the bound state problem with a linear potential in r space. The potential is unbounded at large r leading to a singularity at small q. The singularity is integrable, when regulated by exponentially screening the r-space potential, and is removed by a subtraction technique. The limit of zero screening is taken analytically, and the numerical solution of the subtracted integral equation gives eigenvalues and wave functions in good agreement with position space calculations.

Non-linear vibrations of sandwich viscoelastic shells

NASA Astrophysics Data System (ADS)

Benchouaf, Lahcen; Boutyour, El Hassan; Daya, El Mostafa; Potier-Ferry, Michel

2018-04-01

This paper deals with the non-linear vibration of sandwich viscoelastic shell structures. Coupling a harmonic balance method with the Galerkin's procedure, one obtains an amplitude equation depending on two complex coefficients. The latter are determined by solving a classical eigenvalue problem and two linear ones. This permits to get the non-linear frequency and the non-linear loss factor as functions of the displacement amplitude. To validate our approach, these relationships are illustrated in the case of a circular sandwich ring.

Spin 0 and spin 1/2 quantum relativistic particles in a constant gravitational field

NASA Astrophysics Data System (ADS)

Khorrami, M.; Alimohammadi, M.; Shariati, A.

2003-04-01

The Klein-Gordon and Dirac equations in a semi-infinite lab ( x>0), in the background metric d s2= u2( x)(-d t2+d x2)+d y2+d z2, are investigated. The resulting equations are studied for the special case u( x)=1+ gx. It is shown that in the case of zero transverse-momentum, the square of the energy eigenvalues of the spin-1/2 particles are less than the squares of the corresponding eigenvalues of spin-0 particles with same masses, by an amount of mgℏ c. Finally, for non-zero transverse-momentum, the energy eigenvalues corresponding to large quantum numbers are obtained and the results for spin-0 and spin-1/2 particles are compared to each other.

Iterative Methods for Elliptic Problems and the Discovery of ’q’.

DTIC Science & Technology

1984-07-01

K = M’IlN LN 12 is a nonnegative irreducible matrix. Hence the Perron - Frobenius theory [19] tells us that there is exactly one eigenvalue A with W = p...earlier, the Perron - Frobenius theory implies that p is itself an eigenvalue. However, as we have said, in this instance the eigenvalue problem (l.12a

Goal-based h-adaptivity of the 1-D diamond difference discrete ordinate method

NASA Astrophysics Data System (ADS)

Jeffers, R. S.; Kópházi, J.; Eaton, M. D.; Févotte, F.; Hülsemann, F.; Ragusa, J.

2017-04-01

The quantity of interest (QoI) associated with a solution of a partial differential equation (PDE) is not, in general, the solution itself, but a functional of the solution. Dual weighted residual (DWR) error estimators are one way of providing an estimate of the error in the QoI resulting from the discretisation of the PDE. This paper aims to provide an estimate of the error in the QoI due to the spatial discretisation, where the discretisation scheme being used is the diamond difference (DD) method in space and discrete ordinate (SN) method in angle. The QoI are reaction rates in detectors and the value of the eigenvalue (Keff) for 1-D fixed source and eigenvalue (Keff criticality) neutron transport problems respectively. Local values of the DWR over individual cells are used as error indicators for goal-based mesh refinement, which aims to give an optimal mesh for a given QoI.

Moving mode shape function approach for spinning disk and asymmetric disc brake squeal

NASA Astrophysics Data System (ADS)

Kang, Jaeyoung

2018-06-01

The solution approach of an asymmetric spinning disk under stationary friction loads requires the mode shape function fixed in the disk in the assumed mode method when the equations of motion is described in the space-fixed frame. This model description will be termed the 'moving mode shape function approach' and it allows us to formulate the stationary contact load problem in both the axisymmetric and asymmetric disk cases. Numerical results show that the eigenvalues of the time-periodic axisymmetric disk system are time-invariant. When the axisymmetry of the disk is broken, the positive real parts of the eigenvalues highly vary with the rotation of the disk in the slow speeds in such application as disc brake squeal. By using the Floquet stability analysis, it is also shown that breaking the axisymmetry of the disc alters the stability boundaries of the system.

Distorting general relativity: gravity's rainbow and f(R) theories at work

DOE Office of Scientific and Technical Information (OSTI.GOV)

Garattini, Remo, E-mail: Remo.Garattini@unibg.it

2013-06-01

We compute the Zero Point Energy in a spherically symmetric background combining the high energy distortion of Gravity's Rainbow with the modification induced by a f(R) theory. Here f(R) is a generic analytic function of the Ricci curvature scalar R in 4D and in 3D. The explicit calculation is performed for a Schwarzschild metric. Due to the spherically symmetric property of the Schwarzschild metric we can compare the effects of the modification induced by a f(R) theory in 4D and in 3D. We find that the final effect of the combined theory is to have finite quantities that shift themore » Zero Point Energy. In this context we setup a Sturm-Liouville problem with the cosmological constant considered as the associated eigenvalue. The eigenvalue equation is a reformulation of the Wheeler-DeWitt equation which is analyzed by means of a variational approach based on gaussian trial functionals. With the help of a canonical decomposition, we find that the relevant contribution to one loop is given by the graviton quantum fluctuations around the given background. A final discussion on the connection of our result with the observed cosmological constant is also reported.« less

An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy

NASA Astrophysics Data System (ADS)

Matsushima, Masatomo; Ohmiya, Mayumi

2009-09-01

The stationary KdV hierarchy is constructed using a kind of recursion operator called Λ-operator. The notion of the maximal solution of the n-th stationary KdV equation is introduced. Using this maximal solution, a specific differential polynomial with the auxiliary spectral parameter called the spectral M-function is constructed as the quadratic form of the fundamental system of the eigenvalue problem for the 2-nd order linear ordinary differential equation which is related to the linearizing operator of the hierarchy. By calculating a perfect square condition of the quadratic form by an elementary algebraic method, the complete set of first integrals of this hierarchy is constructed.

The Application of a Boundary Integral Equation Method to the Prediction of Ducted Fan Engine Noise

NASA Technical Reports Server (NTRS)

Dunn, M. H.; Tweed, J.; Farassat, F.

1999-01-01

The prediction of ducted fan engine noise using a boundary integral equation method (BIEM) is considered. Governing equations for the BIEM are based on linearized acoustics and describe the scattering of incident sound by a thin, finite-length cylindrical duct in the presence of a uniform axial inflow. A classical boundary value problem (BVP) is derived that includes an axisymmetric, locally reacting liner on the duct interior. Using potential theory, the BVP is recast as a system of hypersingular boundary integral equations with subsidiary conditions. We describe the integral equation derivation and solution procedure in detail. The development of the computationally efficient ducted fan noise prediction program TBIEM3D, which implements the BIEM, and its utility in conducting parametric noise reduction studies are discussed. Unlike prediction methods based on spinning mode eigenfunction expansions, the BIEM does not require the decomposition of the interior acoustic field into its radial and axial components which, for the liner case, avoids the solution of a difficult complex eigenvalue problem. Numerical spectral studies are presented to illustrate the nexus between the eigenfunction expansion representation and BIEM results. We demonstrate BIEM liner capability by examining radiation patterns for several cases of practical interest.

Eigensolutions of nonviscously damped systems based on the fixed-point iteration

NASA Astrophysics Data System (ADS)

Lázaro, Mario

2018-03-01

In this paper, nonviscous, nonproportional, symmetric vibrating structures are considered. Nonviscously damped systems present dissipative forces depending on the time history of the response via kernel hereditary functions. Solutions of the free motion equation leads to a nonlinear eigenvalue problem involving mass, stiffness and damping matrices, this latter as dependent on frequency. Viscous damping can be considered as a particular case, involving damping forces as function of the instantaneous velocity of the degrees of freedom. In this work, a new numerical procedure to compute eigensolutions is proposed. The method is based on the construction of certain recursive functions which, under a iterative scheme, allow to reach eigenvalues and eigenvectors simultaneously and avoiding computation of eigensensitivities. Eigenvalues can be read then as fixed-points of those functions. A deep analysis of the convergence is carried out, focusing specially on relating the convergence conditions and error-decay rate to the damping model features, such as the nonproportionality and the viscoelasticity. The method is validated using two 6 degrees of freedom numerical examples involving both nonviscous and viscous damping and a continuous system with a local nonviscous damper. The convergence and the sequences behavior are in agreement with the results foreseen by the theory.

Statistical dynamo theory: Mode excitation.

PubMed

Hoyng, P

2009-04-01

We compute statistical properties of the lowest-order multipole coefficients of the magnetic field generated by a dynamo of arbitrary shape. To this end we expand the field in a complete biorthogonal set of base functions, viz. B= summation operator_{k}a;{k}(t)b;{k}(r) . The properties of these biorthogonal function sets are treated in detail. We consider a linear problem and the statistical properties of the fluid flow are supposed to be given. The turbulent convection may have an arbitrary distribution of spatial scales. The time evolution of the expansion coefficients a;{k} is governed by a stochastic differential equation from which we infer their averages a;{k} , autocorrelation functions a;{k}(t)a;{k *}(t+tau) , and an equation for the cross correlations a;{k}a;{l *} . The eigenfunctions of the dynamo equation (with eigenvalues lambda_{k} ) turn out to be a preferred set in terms of which our results assume their simplest form. The magnetic field of the dynamo is shown to consist of transiently excited eigenmodes whose frequency and coherence time is given by Ilambda_{k} and -1/Rlambda_{k} , respectively. The relative rms excitation level of the eigenmodes, and hence the distribution of magnetic energy over spatial scales, is determined by linear theory. An expression is derived for |a;{k}|;{2}/|a;{0}|;{2} in case the fundamental mode b;{0} has a dominant amplitude, and we outline how this expression may be evaluated. It is estimated that |a;{k}|;{2}/|a;{0}|;{2} approximately 1/N , where N is the number of convective cells in the dynamo. We show that the old problem of a short correlation time (or first-order smoothing approximation) has been partially eliminated. Finally we prove that for a simple statistically steady dynamo with finite resistivity all eigenvalues obey Rlambda_{k}<0 .

The Kirkwood{endash}Buckingham variational method and the boundary value problems for the molecular Schr{umlt o}dinger equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pupyshev, V.I.; Scherbinin, A.V.; Stepanov, N.F.

1997-11-01

The approach based on the multiplicative form of a trial wave function within the framework of the variational method, initially proposed by Kirkwood and Buckingham, is shown to be an effective analytical tool in the quantum mechanical study of atoms and molecules. As an example, the elementary proof is given to the fact that the ground state energy of a molecular system placed into the box with walls of finite height goes to the corresponding eigenvalue of the Dirichlet boundary value problem when the height of the walls is growing up to infinity. {copyright} {ital 1997 American Institute of Physics.}

Investigation, development and application of optimal output feedback theory. Vol. 4: Measures of eigenvalue/eigenvector sensitivity to system parameters and unmodeled dynamics

NASA Technical Reports Server (NTRS)

Halyo, Nesim

1987-01-01

Some measures of eigenvalue and eigenvector sensitivity applicable to both continuous and discrete linear systems are developed and investigated. An infinite series representation is developed for the eigenvalues and eigenvectors of a system. The coefficients of the series are coupled, but can be obtained recursively using a nonlinear coupled vector difference equation. A new sensitivity measure is developed by considering the effects of unmodeled dynamics. It is shown that the sensitivity is high when any unmodeled eigenvalue is near a modeled eigenvalue. Using a simple example where the sensor dynamics have been neglected, it is shown that high feedback gains produce high eigenvalue/eigenvector sensitivity. The smallest singular value of the return difference is shown not to reflect eigenvalue sensitivity since it increases with the feedback gains. Using an upper bound obtained from the infinite series, a procedure to evaluate whether the sensitivity to parameter variations is within given acceptable bounds is developed and demonstrated by an example.

PT-symmetric eigenvalues for homogeneous potentials

NASA Astrophysics Data System (ADS)

Eremenko, Alexandre; Gabrielov, Andrei

2018-05-01

We consider one-dimensional Schrödinger equations with potential x2M(ix)ɛ, where M ≥ 1 is an integer and ɛ is real, under appropriate parity and time (PT)-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as ɛ changes, the real spectrum suddenly becomes non-real in the sense that all but finitely many eigenvalues become non-real. We find the limit arguments of these non-real eigenvalues E as E → ∞.

Soliton's eigenvalue based analysis on the generation mechanism of rogue wave phenomenon in optical fibers exhibiting weak third order dispersion.

PubMed

Weerasekara, Gihan; Tokunaga, Akihiro; Terauchi, Hiroki; Eberhard, Marc; Maruta, Akihiro

2015-01-12

One of the extraordinary aspects of nonlinear wave evolution which has been observed as the spontaneous occurrence of astonishing and statistically extraordinary amplitude wave is called rogue wave. We show that the eigenvalues of the associated equation of nonlinear Schrödinger equation are almost constant in the vicinity of rogue wave and we validate that optical rogue waves are formed by the collision between quasi-solitons in anomalous dispersion fiber exhibiting weak third order dispersion.

A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields

DOE PAGES

Osborn, Sarah; Vassilevski, Panayot S.; Villa, Umberto

2017-10-26

In this paper, we propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen--Loève (KL) decomposition. However, the KL expansion requires solving a dense eigenvalue problem and is therefore computationally infeasible for large-scale problems. Sampling methods based on stochastic partial differential equations provide a highly scalable way to sample Gaussian fields, but the resulting parametrization is mesh dependent. We propose a multilevel decomposition of the stochastic field to allow for scalable, hierarchical sampling based on solving amore » mixed finite element formulation of a stochastic reaction-diffusion equation with a random, white noise source function. Lastly, numerical experiments are presented to demonstrate the scalability of the sampling method as well as numerical results of multilevel Monte Carlo simulations for a subsurface porous media flow application using the proposed sampling method.« less

A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields

DOE Office of Scientific and Technical Information (OSTI.GOV)

Osborn, Sarah; Vassilevski, Panayot S.; Villa, Umberto

In this paper, we propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen--Loève (KL) decomposition. However, the KL expansion requires solving a dense eigenvalue problem and is therefore computationally infeasible for large-scale problems. Sampling methods based on stochastic partial differential equations provide a highly scalable way to sample Gaussian fields, but the resulting parametrization is mesh dependent. We propose a multilevel decomposition of the stochastic field to allow for scalable, hierarchical sampling based on solving amore » mixed finite element formulation of a stochastic reaction-diffusion equation with a random, white noise source function. Lastly, numerical experiments are presented to demonstrate the scalability of the sampling method as well as numerical results of multilevel Monte Carlo simulations for a subsurface porous media flow application using the proposed sampling method.« less

Optical potential approach to the electron-atom impact ionization threshold problem

NASA Technical Reports Server (NTRS)

Temkin, A.; Hahn, Y.

1973-01-01

The problem of the threshold law for electron-atom impact ionization is reconsidered as an extrapolation of inelastic cross sections through the ionization threshold. The cross sections are evaluated from a distorted wave matrix element, the final state of which describes the scattering from the Nth excited state of the target atom. The actual calculation is carried for the e-H system, and a model is introduced which is shown to preserve the essential properties of the problem while at the same time reducing the dimensionability of the Schrodinger equation. Nevertheless, the scattering equation is still very complex. It is dominated by the optical potential which is expanded in terms of eigen-spectrum of QHQ. It is shown by actual calculation that the lower eigenvalues of this spectrum descend below the relevant inelastic thresholds; it follows rigorously that the optical potential contains repulsive terms. Analytical solutions of the final state wave function are obtained with several approximations of the optical potential.

Kinetic theory of nonlinear diffusion in a weakly disordered nonlinear Schrödinger chain in the regime of homogeneous chaos.

PubMed

Basko, D M

2014-02-01

We study the discrete nonlinear Schröinger equation with weak disorder, focusing on the regime when the nonlinearity is, on the one hand, weak enough for the normal modes of the linear problem to remain well resolved but, on the other, strong enough for the dynamics of the normal mode amplitudes to be chaotic for almost all modes. We show that in this regime and in the limit of high temperature, the macroscopic density ρ satisfies the nonlinear diffusion equation with a density-dependent diffusion coefficient, D(ρ) = D(0)ρ(2). An explicit expression for D(0) is obtained in terms of the eigenfunctions and eigenvalues of the linear problem, which is then evaluated numerically. The role of the second conserved quantity (energy) in the transport is also quantitatively discussed.

A wide-angle high Mach number modal expansion for infrasound propagation.

PubMed

Assink, Jelle; Waxler, Roger; Velea, Doru

2017-03-01

The use of modal expansions to solve the problem of atmospheric infrasound propagation is revisited. A different form of the associated modal equation is introduced, valid for wide-angle propagation in atmospheres with high Mach number flow. The modal equation can be formulated as a quadratic eigenvalue problem for which there are simple and efficient numerical implementations. A perturbation expansion for the treatment of attenuation, valid for stratified media with background flow, is derived as well. Comparisons are carried out between the proposed algorithm and a modal algorithm assuming an effective sound speed, including a real data case study. The comparisons show that the effective sound speed approximation overestimates the effect of horizontal wind on sound propagation, leading to errors in traveltime, propagation path, trace velocity, and absorption. The error is found to be dependent on propagation angle and Mach number.

Global stability analysis of axisymmetric boundary layer over a circular cylinder

NASA Astrophysics Data System (ADS)

Bhoraniya, Ramesh; Vinod, Narayanan

2018-05-01

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier-Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi's iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.

The Cr dependence problem of eigenvalues of the Laplace operator on domains in the plane

NASA Astrophysics Data System (ADS)

Haddad, Julian; Montenegro, Marcos

2018-03-01

The Cr dependence problem of multiple Dirichlet eigenvalues on domains is discussed for elliptic operators by regarding C r + 1-smooth one-parameter families of C1 perturbations of domains in Rn. As applications of our main theorem (Theorem 1), we provide a fairly complete description for all eigenvalues of the Laplace operator on disks and squares in R2 and also for its second eigenvalue on balls in Rn for any n ≥ 3. The central tool used in our proof is a degenerate implicit function theorem on Banach spaces (Theorem 2) of independent interest.

An Isoperimetric Inequality for Fundamental Tones of Free Plates

ERIC Educational Resources Information Center

Chasman, Laura

2009-01-01

We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given tau greater than 0, the free plate eigenvalues omega and eigenfunctions upsilon are determined by the equation Delta Delta upsilon - tau Delta upsilon = omega upsilon together with…

The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak

NASA Astrophysics Data System (ADS)

Xie, T.; Zhang, Y. Z.; Mahajan, S. M.; Hu, S. L.; He, Hongda; Liu, Z. Y.

2017-10-01

The two-dimensional (2D) kinetic ballooning theory is developed for the ion temperature gradient mode in an up-down symmetric equilibrium (illustrated via concentric circular magnetic surfaces). The ballooning transform converts the basic 2D linear gyro-kinetic equation into two equations: (1) the lowest order equation (ballooning equation) is an integral equation essentially the same as that reported by Dong et al., [Phys. Fluids B 4, 1867 (1992)] but has an undetermined Floquet phase variable, (2) the higher order equation for the rapid phase envelope is an ordinary differential equation in the same form as the 2D ballooning theory in a fluid model [Xie et al., Phys. Plasmas 23, 042514 (2016)]. The system is numerically solved by an iterative approach to obtain the (phase independent) eigen-value. The new results are compared to the two earlier theories. We find a strongly modified up-down asymmetric mode structure, and non-trivial modifications to the eigen-value.

Effect of curvature on stationary crossflow instability of a three-dimensional boundary layer

NASA Technical Reports Server (NTRS)

Lin, Ray-Sing; Reed, Helen L.

1993-01-01

An incompressible three-dimensional laminar boundary-layer flow over a swept wing is used as a model to study both the wall-curvature and streamline-curvature effects on the stationary crossflow instability. The basic state is obtained by solving the full Navier-Stokes (N-S) equations numerically. The linear disturbance equations are cast on a fixed, body-intrinsic, curvilinear coordinate system. Those nonparallel terms which contribute mainly to the streamline-curvature effect are retained in the formulation of the disturbance equations and approximated by their local finite difference values. The resulting eigenvalue problem is solved by a Chebyshev collocation method. The present results indicate that the convex wall curvature has a stabilizing effect, whereas the streamline curvature has a destabilizing effect. A validation of these effects with an N-S solution for the linear disturbance flow is provided.

Non-perturbative background field calculations

NASA Astrophysics Data System (ADS)

Stephens, C. R.

1988-01-01

New methods are developed for calculating one loop functional determinants in quantum field theory. Instead of relying on a calculation of all the eigenvalues of the small fluctuation equation, these techniques exploit the ability of the proper time formalism to reformulate an infinite dimensional field theoretic problem into a finite dimensional covariant quantum mechanical analog, thereby allowing powerful tools such as the method of Jacobi fields to be used advantageously in a field theory setting. More generally the methods developed herein should be extremely valuable when calculating quantum processes in non-constant background fields, offering a utilitarian alternative to the two standard methods of calculation—perturbation theory in the background field or taking the background field into account exactly. The formalism developed also allows for the approximate calculation of covariances of partial differential equations from a knowledge of the solutions of a homogeneous ordinary differential equation.

Benchmark problems in computational aeroacoustics

NASA Technical Reports Server (NTRS)

Porter-Locklear, Freda

1994-01-01

A recent directive at NASA Langley is aimed at numerically predicting principal noise sources. During my summer stay, I worked with high-order ENO code, developed by Dr. Harold Atkins, for solving the unsteady compressible Navier-Stokes equations, as it applies to computational aeroacoustics (CAA). A CAA workshop, composed of six categories of benchmark problems, has been organized to test various numerical properties of code. My task was to determine the robustness of Atkins' code for these test problems. In one category, we tested the nonlinear wave propagation of the code for the one-dimensional Euler equations, with initial pressure, density, and velocity conditions. Using freestream boundary conditions, our results were plausible. In another category, we solved the linearized two-dimensional Euler equations to test the effectiveness of radiation boundary conditions. Here we utilized MAPLE to compute eigenvalues and eigenvectors of the Jacobian given variable and flux vectors. We experienced a minor problem with inflow and outflow boundary conditions. Next, we solved the quasi one dimensional unsteady flow equations with an incoming acoustic wave of amplitude 10(exp -6). The small amplitude sound wave was incident on a convergent-divergent nozzle. After finding a steady-state solution and then marching forward, our solution indicated that after 30 periods the acoustic wave had dissipated (a period is time required for sound wave to traverse one end of nozzle to other end).

Extension of the tridiagonal reduction (FEER) method for complex eigenvalue problems in NASTRAN

NASA Technical Reports Server (NTRS)

Newman, M.; Mann, F. I.

1978-01-01

As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum were extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order was much lower than that of the full size problem. The reduction process was effected automatically, and thus avoided the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admitted mass, damping, and stiffness matrices which were unrestricted in character, i.e., they might be real, symmetric or nonsymmetric, singular or nonsingular.

Properties of coupled-cluster equations originating in excitation sub-algebras

NASA Astrophysics Data System (ADS)

Kowalski, Karol

2018-03-01

In this paper, we discuss properties of single-reference coupled cluster (CC) equations associated with the existence of sub-algebras of excitations that allow one to represent CC equations in a hybrid fashion where the cluster amplitudes associated with these sub-algebras can be obtained by solving the corresponding eigenvalue problem. For closed-shell formulations analyzed in this paper, the hybrid representation of CC equations provides a natural way for extending active-space and seniority number concepts to provide an accurate description of electron correlation effects. Moreover, a new representation can be utilized to re-define iterative algorithms used to solve CC equations, especially for tough cases defined by the presence of strong static and dynamical correlation effects. We will also explore invariance properties associated with excitation sub-algebras to define a new class of CC approximations referred to in this paper as the sub-algebra-flow-based CC methods. We illustrate the performance of these methods on the example of ground- and excited-state calculations for commonly used small benchmark systems.

Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction.

PubMed

Ratowsky, R P; Fleck, J A

1991-06-01

The Lanczos recursion algorithm is used to determine forward-propagating solutions for both the paraxial and Helmholtz wave equations for longitudinally invariant refractive indices. By eigenvalue analysis it is demonstrated that the method gives extremely accurate solutions to both equations.

A multilevel finite element method for Fredholm integral eigenvalue problems

NASA Astrophysics Data System (ADS)

Xie, Hehu; Zhou, Tao

2015-12-01

In this work, we proposed a multigrid finite element (MFE) method for solving the Fredholm integral eigenvalue problems. The main motivation for such studies is to compute the Karhunen-Loève expansions of random fields, which play an important role in the applications of uncertainty quantification. In our MFE framework, solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh. Then, any existing efficient integration scheme can be used for the associated integration process. The error estimates are provided, and the computational complexity is analyzed. It is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh. Several numerical experiments are presented to validate the efficiency of the proposed numerical method.

A proposed method for enhanced eigen-pair extraction using finite element methods: Theory and application

NASA Technical Reports Server (NTRS)

Jara-Almonte, J.; Mitchell, L. D.

1988-01-01

The paper covers two distinct parts: theory and application. The goal of this work was the reduction of model size with an increase in eigenvalue/vector accuracy. This method is ideal for the condensation of large truss- or beam-type structures. The theoretical approach involves the conversion of a continuum transfer matrix beam element into an 'Exact' dynamic stiffness element. This formulation is implemented in a finite element environment. This results in the need to solve a transcendental eigenvalue problem. Once the eigenvalue is determined the eigenvectors can be reconstructed with any desired spatial precision. No discretization limitations are imposed on the reconstruction. The results of such a combined finite element and transfer matrix formulation is a much smaller FEM eigenvalue problem. This formulation has the ability to extract higher eigenvalues as easily and as accurately as lower eigenvalues. Moreover, one can extract many more eigenvalues/vectors from the model than the number of degrees of freedom in the FEM formulation. Typically, the number of eigenvalues accurately extractable via the 'Exact' element method are at least 8 times the number of degrees of freedom. In contrast, the FEM usually extracts one accurate (within 5 percent) eigenvalue for each 3-4 degrees of freedom. The 'Exact' element results in a 20-30 improvement in the number of accurately extractable eigenvalues and eigenvectors.

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science.

PubMed

Marek, A; Blum, V; Johanni, R; Havu, V; Lang, B; Auckenthaler, T; Heinecke, A; Bungartz, H-J; Lederer, H

2014-05-28

Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem sizes arising in the field of electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a matrix of dimension 260,000, scalability up to 295,000 CPU cores has been shown on BlueGene/P.

Development of an unsteady wake theory appropriate for aeroelastic analyses of rotors in hover and forward flight

NASA Technical Reports Server (NTRS)

Peters, David A.

1988-01-01

The purpose of this research is the development of an unsteady aerodynamic model for rotors such that it can be used in conventional aeroelastic analysis (e.g., eigenvalue determination and control system design). For this to happen, the model must be in a state-space formulation such that the states of the flow can be defined, calculated and identified as part of the analysis. The fluid mechanics of the problem is given by a closed-form inversion of an acceleration potential. The result is a set of first-order differential equations in time for the unknown flow coefficients. These equations are hierarchical in the sense that they may be truncated at any number of radial or azimuthal terms.

The use of an analytic Hamiltonian matrix for solving the hydrogenic atom

NASA Astrophysics Data System (ADS)

Bhatti, Mohammad

2001-10-01

The non-relativistic Hamiltonian corresponding to the Shrodinger equation is converted into analytic Hamiltonian matrix using the kth order B-splines functions. The Galerkin method is applied to the solution of the Shrodinger equation for bound states of hydrogen-like systems. The program Mathematica is used to create analytic matrix elements and exact integration is performed over the knot-sequence of B-splines and the resulting generalized eigenvalue problem is solved on a specified numerical grid. The complete basis set and the energy spectrum is obtained for the coulomb potential for hydrogenic systems with Z less than 100 with B-splines of order eight. Another application is given to test the Thomas-Reiche-Kuhn sum rule for the hydrogenic systems.

Complex eigenvalue extraction in NASTRAN by the tridiagonal reduction (FEER) method

NASA Technical Reports Server (NTRS)

Newman, M.; Mann, F. I.

1977-01-01

An extension of the Tridiagonal Reduction (FEER) method to complex eigenvalue analysis in NASTRAN is described. As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum are extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order is much lower than that of the full size problem. The reduction process is effected automatically, and thus avoids the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admits mass, damping and stiffness matrices which are unrestricted in character, i.e., they may be real, complex, symmetric or unsymmetric, singular or non-singular.

A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting

NASA Technical Reports Server (NTRS)

Shu, Chi-Wang

1992-01-01

The present treatment of elliptic regions via hyperbolic flux-splitting and high order methods proposes a flux splitting in which the corresponding Jacobians have real and positive/negative eigenvalues. While resembling the flux splitting used in hyperbolic systems, the present generalization of such splitting to elliptic regions allows the handling of mixed-type systems in a unified and heuristically stable fashion. The van der Waals fluid-dynamics equation is used. Convergence with good resolution to weak solutions for various Riemann problems are observed.

Crossing symmetry in alpha space

NASA Astrophysics Data System (ADS)

Hogervorst, Matthijs; van Rees, Balt C.

2017-11-01

We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number α. This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form.

Use of asymptotic methods in vibration analysis

NASA Technical Reports Server (NTRS)

Ashley, H.

1978-01-01

The derivation of dynamic differential equations, suitable for studying the vibrations of rotating, curved, slender structures was examined, and the Hamiltonian procedure was advocated for this purpose. Various reductions of the full system are displayed, which govern the vibrating troposkien when various order of magnitude restrictions are placed on important parameters. Possible advantages of the WKB asymptotic method for solving these classes of problems are discussed. A special case of this method is used illustratively to calculate eigenvalues and eigenfunctions for a flat turbine blade with small flexural stiffness.

Wavefunction Engineering of Spintronic devices in ZnO/MgO and GaN/AlN Quantum Structures Doped with Transition Metal Ions

DTIC Science & Technology

2006-08-01

2005). 7. " Dependence of the interband transitions on the In mole-fraction and the applied electric field in InxGaj_xAs/In0. 52Al0.48As multiple... tunneling boundary conditions for open structures. The boundary conditions at interfaces require the maintenance of derivative operator ordering...computational methods for the solution of Schr6dinger’s equations for scattering/ tunneling structures as well as for the eigenvalue problems that arise for

Construction, classification and parametrization of complex Hadamard matrices

NASA Astrophysics Data System (ADS)

Szöllősi, Ferenc

To improve the design of nuclear systems, high-fidelity neutron fluxes are required. Leadership-class machines provide platforms on which very large problems can be solved. Computing such fluxes efficiently requires numerical methods with good convergence properties and algorithms that can scale to hundreds of thousands of cores. Many 3-D deterministic transport codes are decomposable in space and angle only, limiting them to tens of thousands of cores. Most codes rely on methods such as Gauss Seidel for fixed source problems and power iteration for eigenvalue problems, which can be slow to converge for challenging problems like those with highly scattering materials or high dominance ratios. Three methods have been added to the 3-D SN transport code Denovo that are designed to improve convergence and enable the full use of cutting-edge computers. The first is a multigroup Krylov solver that converges more quickly than Gauss Seidel and parallelizes the code in energy such that Denovo can use hundreds of thousand of cores effectively. The second is Rayleigh quotient iteration (RQI), an old method applied in a new context. This eigenvalue solver finds the dominant eigenvalue in a mathematically optimal way and should converge in fewer iterations than power iteration. RQI creates energy-block-dense equations that the new Krylov solver treats efficiently. However, RQI can have convergence problems because it creates poorly conditioned systems. This can be overcome with preconditioning. The third method is a multigrid-in-energy preconditioner. The preconditioner takes advantage of the new energy decomposition because the grids are in energy rather than space or angle. The preconditioner greatly reduces iteration count for many problem types and scales well in energy. It also allows RQI to be successful for problems it could not solve otherwise. The methods added to Denovo accomplish the goals of this work. They converge in fewer iterations than traditional methods and enable the use of hundreds of thousands of cores. Each method can be used individually, with the multigroup Krylov solver and multigrid-in-energy preconditioner being particularly successful on their own. The largest benefit, though, comes from using these methods in concert.

Analytical bound-state solutions of the Schrödinger equation for the Manning-Rosen plus Hulthén potential within SUSY quantum mechanics

NASA Astrophysics Data System (ADS)

Ahmadov, A. I.; Naeem, Maria; Qocayeva, M. V.; Tarverdiyeva, V. A.

2018-01-01

In this paper, the bound-state solution of the modified radial Schrödinger equation is obtained for the Manning-Rosen plus Hulthén potential by using new developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial wave functions are defined for any l≠0 angular momentum case via the Nikiforov-Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. Thanks to both methods, equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is presented. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary l states. A closed form of the normalization constant of the wave functions is also found. It is shown that, the energy eigenvalues and eigenfunctions are sensitive to nr radial and l orbital quantum numbers.

A Brief Historical Introduction to Determinants with Applications

ERIC Educational Resources Information Center

Debnath, L.

2013-01-01

This article deals with a short historical introduction to determinants with applications to the theory of equations, geometry, multiple integrals, differential equations and linear algebra. Included are some properties of determinants with proofs, eigenvalues, eigenvectors and characteristic equations with examples of applications to simple…

Laplace and Z Transform Solutions of Differential and Difference Equations With the HP-41C.

ERIC Educational Resources Information Center

Harden, Richard C.; Simons, Fred O., Jr.

1983-01-01

A previously developed program for the HP-41C programmable calculator is extended to handle models of differential and difference equations with multiple eigenvalues. How to obtain difference equation solutions via the Z transform is described. (MNS)

Rapid solution of large-scale systems of equations

NASA Technical Reports Server (NTRS)

Storaasli, Olaf O.

1994-01-01

The analysis and design of complex aerospace structures requires the rapid solution of large systems of linear and nonlinear equations, eigenvalue extraction for buckling, vibration and flutter modes, structural optimization and design sensitivity calculation. Computers with multiple processors and vector capabilities can offer substantial computational advantages over traditional scalar computer for these analyses. These computers fall into two categories: shared memory computers and distributed memory computers. This presentation covers general-purpose, highly efficient algorithms for generation/assembly or element matrices, solution of systems of linear and nonlinear equations, eigenvalue and design sensitivity analysis and optimization. All algorithms are coded in FORTRAN for shared memory computers and many are adapted to distributed memory computers. The capability and numerical performance of these algorithms will be addressed.

Filter method without boundary-value condition for simultaneous calculation of eigenfunction and eigenvalue of a stationary Schrödinger equation on a grid.

PubMed

Nurhuda, M; Rouf, A

2017-09-01

The paper presents a method for simultaneous computation of eigenfunction and eigenvalue of the stationary Schrödinger equation on a grid, without imposing boundary-value condition. The method is based on the filter operator, which selects the eigenfunction from wave packet at the rate comparable to δ function. The efficacy and reliability of the method are demonstrated by comparing the simulation results with analytical or numerical solutions obtained by using other methods for various boundary-value conditions. It is found that the method is robust, accurate, and reliable. Further prospect of filter method for simulation of the Schrödinger equation in higher-dimensional space will also be highlighted.

Thermo-magnetic field effects on the wave propagation behavior of smart magnetostrictive sandwich nanoplates

NASA Astrophysics Data System (ADS)

Ebrahimi, Farzad; Dabbagh, Ali

2018-03-01

In this paper, a three-variable plate model is utilized to explore the wave propagation problem of smart sandwich nanoplates made of a magnetostrictive core and ceramic face sheets while subjected to thermo-magnetic loading. Herein, the magnetostriction effect is considered and controlled via a feedback control system. The nanoplate is supposed to be embedded on a visco-Pasternak elastic substrate. The kinematic relations are derived based on the Kirchhoff plate theory; also, combining these obtained equations with Hamilton's principle, the local equations of motion are achieved. According to a nonlocal strain gradient theory (NSGT), the small-scale influences are covered precisely by introducing two scale coefficients. Afterwards, the nonlocal governing equations are derived coupling the local equations with those of the NSGT. Applying an analytical solution, the wave frequency and phase velocity of the propagated waves can be gathered solving an eigenvalue problem. On the other hand, accuracy and efficiency of the presented model are verified by setting a comparison between the obtained results with those of previous published researches. Effects of different variants are plotted in some figures and the highlights are discussed in detail.

Multiscale modeling and computation of optically manipulated nano devices

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bao, Gang, E-mail: baog@zju.edu.cn; Liu, Di, E-mail: richardl@math.msu.edu; Luo, Songting, E-mail: luos@iastate.edu

2016-07-01

We present a multiscale modeling and computational scheme for optical-mechanical responses of nanostructures. The multi-physical nature of the problem is a result of the interaction between the electromagnetic (EM) field, the molecular motion, and the electronic excitation. To balance accuracy and complexity, we adopt the semi-classical approach that the EM field is described classically by the Maxwell equations, and the charged particles follow the Schrödinger equations quantum mechanically. To overcome the numerical challenge of solving the high dimensional multi-component many-body Schrödinger equations, we further simplify the model with the Ehrenfest molecular dynamics to determine the motion of the nuclei, andmore » use the Time-Dependent Current Density Functional Theory (TD-CDFT) to calculate the excitation of the electrons. This leads to a system of coupled equations that computes the electromagnetic field, the nuclear positions, and the electronic current and charge densities simultaneously. In the regime of linear responses, the resonant frequencies initiating the out-of-equilibrium optical-mechanical responses can be formulated as an eigenvalue problem. A self-consistent multiscale method is designed to deal with the well separated space scales. The isomerization of azobenzene is presented as a numerical example.« less

Convergence analysis of two-node CMFD method for two-group neutron diffusion eigenvalue problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jeong, Yongjin; Park, Jinsu; Lee, Hyun Chul

2015-12-01

In this paper, the nonlinear coarse-mesh finite difference method with two-node local problem (CMFD2N) is proven to be unconditionally stable for neutron diffusion eigenvalue problems. The explicit current correction factor (CCF) is derived based on the two-node analytic nodal method (ANM2N), and a Fourier stability analysis is applied to the linearized algorithm. It is shown that the analytic convergence rate obtained by the Fourier analysis compares very well with the numerically measured convergence rate. It is also shown that the theoretical convergence rate is only governed by the converged second harmonic buckling and the mesh size. It is also notedmore » that the convergence rate of the CCF of the CMFD2N algorithm is dependent on the mesh size, but not on the total problem size. This is contrary to expectation for eigenvalue problem. The novel points of this paper are the analytical derivation of the convergence rate of the CMFD2N algorithm for eigenvalue problem, and the convergence analysis based on the analytic derivations.« less

Asymptotic theory of a slender rotating beam with end masses.

NASA Technical Reports Server (NTRS)

Whitman, A. M.; Abel, J. M.

1972-01-01

The method of matched asymptotic expansions is employed to solve the singular perturbation problem of the vibrations of a rotating beam of small flexural rigidity with concentrated end masses. The problem is complicated by the appearance of the eigenvalue in the boundary conditions. Eigenfunctions and eigenvalues are developed as power series in the perturbation parameter beta to the 1/2 power, and results are given for mode shapes and eigenvalues through terms of the order of beta.

Solution of the symmetric eigenproblem AX=lambda BX by delayed division

NASA Technical Reports Server (NTRS)

Thurston, G. A.; Bains, N. J. C.

1986-01-01

Delayed division is an iterative method for solving the linear eigenvalue problem AX = lambda BX for a limited number of small eigenvalues and their corresponding eigenvectors. The distinctive feature of the method is the reduction of the problem to an approximate triangular form by systematically dropping quadratic terms in the eigenvalue lambda. The report describes the pivoting strategy in the reduction and the method for preserving symmetry in submatrices at each reduction step. Along with the approximate triangular reduction, the report extends some techniques used in the method of inverse subspace iteration. Examples are included for problems of varying complexity.

Oscilaciones estelares no-radiales: aplicación a configuraciones politrópicas y modelos de enanas blancas de He

NASA Astrophysics Data System (ADS)

Córsico, A. H.; Benvenuto, O. G.

Recently in our Observatory we have developed a new Stellar Pulsation Code, independently of other workers. Such program computes eigenvalues (eigenfrequencies) and eigenfunctions of non-radial modes in spherical non-perturbated stellar models. To accomplish this calculations, the four order eigenvalue problem (in the linear adiabatic approach) is solved by means of the well-know technique of Henyey on the finite differences scheme wich replace to the differential equations of the problem. In order to test the Code, we have computed numerous eigenmodes in polytropic configurations for several values of index n. In this comunication we show the excelent agreement of our results and that best available in the literature. Also, we present results of oscillations in models of white dwarf stars with homogeneus chemical composition (pure Helium). This models have been obtained with the Evolution Stellar Code of our Observatory. The calculations outlined above conform a first preliminary step in a major proyect whose main purpose is the study of pulsational properties of DA, DB and DO white dwarfs stars. Detailed investigations have demonstrated that such objets pulsates in non-radial g-modes with eigenperiods in the range 100-2000 sec.

An analytical method for free vibration analysis of functionally graded beams with edge cracks

NASA Astrophysics Data System (ADS)

Wei, Dong; Liu, Yinghua; Xiang, Zhihai

2012-03-01

In this paper, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation. The governing differential equations of motion for an FGM beam are established and the corresponding solutions are found first. The discontinuity of rotation caused by the cracks is simulated by means of the rotational spring model. Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams. The main advantage of the proposed method is that the eigenvalue equation for vibrating beams with an arbitrary number of cracks can be conveniently determined from a third-order determinant. Due to the decrease in the determinant order as compared with previous methods, the developed method is simpler and more convenient to analytically solve the free vibration problem of cracked FGM beams. Moreover, free vibration analyses of the Euler-Bernoulli and Timoshenko beams with any number of cracks can be conducted using the unified procedure based on the developed method. These advantages of the proposed procedure would be more remarkable as the increase of the number of cracks. A comprehensive analysis is conducted to investigate the influences of the location and total number of cracks, material properties, axial load, inertia and end supports on the natural frequencies and vibration mode shapes of FGM beams. The present work may be useful for the design and control of damaged structures.

Towards adjoint-based inversion for rheological parameters in nonlinear viscous mantle flow

NASA Astrophysics Data System (ADS)

Worthen, Jennifer; Stadler, Georg; Petra, Noemi; Gurnis, Michael; Ghattas, Omar

2014-09-01

We address the problem of inferring mantle rheological parameter fields from surface velocity observations and instantaneous nonlinear mantle flow models. We formulate this inverse problem as an infinite-dimensional nonlinear least squares optimization problem governed by nonlinear Stokes equations. We provide expressions for the gradient of the cost functional of this optimization problem with respect to two spatially-varying rheological parameter fields: the viscosity prefactor and the exponent of the second invariant of the strain rate tensor. Adjoint (linearized) Stokes equations, which are characterized by a 4th order anisotropic viscosity tensor, facilitates efficient computation of the gradient. A quasi-Newton method for the solution of this optimization problem is presented, which requires the repeated solution of both nonlinear forward Stokes and linearized adjoint Stokes equations. For the solution of the nonlinear Stokes equations, we find that Newton’s method is significantly more efficient than a Picard fixed point method. Spectral analysis of the inverse operator given by the Hessian of the optimization problem reveals that the numerical eigenvalues collapse rapidly to zero, suggesting a high degree of ill-posedness of the inverse problem. To overcome this ill-posedness, we employ Tikhonov regularization (favoring smooth parameter fields) or total variation (TV) regularization (favoring piecewise-smooth parameter fields). Solution of two- and three-dimensional finite element-based model inverse problems show that a constant parameter in the constitutive law can be recovered well from surface velocity observations. Inverting for a spatially-varying parameter field leads to its reasonable recovery, in particular close to the surface. When inferring two spatially varying parameter fields, only an effective viscosity field and the total viscous dissipation are recoverable. Finally, a model of a subducting plate shows that a localized weak zone at the plate boundary can be partially recovered, especially with TV regularization.

Convergence of discrete Aubry–Mather model in the continuous limit

NASA Astrophysics Data System (ADS)

Su, Xifeng; Thieullen, Philippe

2018-05-01

We develop two approximation schemes for solving the cell equation and the discounted cell equation using Aubry–Mather–Fathi theory. The Hamiltonian is supposed to be Tonelli, time-independent and periodic in space. By Legendre transform it is equivalent to find a fixed point of some nonlinear operator, called Lax-Oleinik operator, which may be discounted or not. By discretizing in time, we are led to solve an additive eigenvalue problem involving a discrete Lax–Oleinik operator. We show how to approximate the effective Hamiltonian and some weak KAM solutions by letting the time step in the discrete model tend to zero. We also obtain a selected discrete weak KAM solution as in Davini et al (2016 Invent. Math. 206 29–55), and show that it converges to a particular solution of the cell equation. In order to unify the two settings, continuous and discrete, we develop a more general formalism of the short-range interactions.

Cucheb: A GPU implementation of the filtered Lanczos procedure

NASA Astrophysics Data System (ADS)

Aurentz, Jared L.; Kalantzis, Vassilis; Saad, Yousef

2017-11-01

This paper describes the software package Cucheb, a GPU implementation of the filtered Lanczos procedure for the solution of large sparse symmetric eigenvalue problems. The filtered Lanczos procedure uses a carefully chosen polynomial spectral transformation to accelerate convergence of the Lanczos method when computing eigenvalues within a desired interval. This method has proven particularly effective for eigenvalue problems that arise in electronic structure calculations and density functional theory. We compare our implementation against an equivalent CPU implementation and show that using the GPU can reduce the computation time by more than a factor of 10. Program Summary Program title: Cucheb Program Files doi:http://dx.doi.org/10.17632/rjr9tzchmh.1 Licensing provisions: MIT Programming language: CUDA C/C++ Nature of problem: Electronic structure calculations require the computation of all eigenvalue-eigenvector pairs of a symmetric matrix that lie inside a user-defined real interval. Solution method: To compute all the eigenvalues within a given interval a polynomial spectral transformation is constructed that maps the desired eigenvalues of the original matrix to the exterior of the spectrum of the transformed matrix. The Lanczos method is then used to compute the desired eigenvectors of the transformed matrix, which are then used to recover the desired eigenvalues of the original matrix. The bulk of the operations are executed in parallel using a graphics processing unit (GPU). Runtime: Variable, depending on the number of eigenvalues sought and the size and sparsity of the matrix. Additional comments: Cucheb is compatible with CUDA Toolkit v7.0 or greater.

Sensitivity analysis of eigenvalues for an electro-hydraulic servomechanism

NASA Astrophysics Data System (ADS)

Stoia-Djeska, M.; Safta, C. A.; Halanay, A.; Petrescu, C.

2012-11-01

Electro-hydraulic servomechanisms (EHSM) are important components of flight control systems and their role is to control the movement of the flying control surfaces in response to the movement of the cockpit controls. As flight-control systems, the EHSMs have a fast dynamic response, a high power to inertia ratio and high control accuracy. The paper is devoted to the study of the sensitivity for an electro-hydraulic servomechanism used for an aircraft aileron action. The mathematical model of the EHSM used in this paper includes a large number of parameters whose actual values may vary within some ranges of uncertainty. It consists in a nonlinear ordinary differential equation system composed by the mass and energy conservation equations, the actuator movement equations and the controller equation. In this work the focus is on the sensitivities of the eigenvalues of the linearized homogeneous system, which are the partial derivatives of the eigenvalues of the state-space system with respect the parameters. These are obtained using a modal approach based on the eigenvectors of the state-space direct and adjoint systems. To calculate the eigenvalues and their sensitivity the system's Jacobian and its partial derivatives with respect the parameters are determined. The calculation of the derivative of the Jacobian matrix with respect to the parameters is not a simple task and for many situations it must be done numerically. The system stability is studied in relation with three parameters: m, the equivalent inertial load of primary control surface reduced to the actuator rod; B, the bulk modulus of oil and p a pressure supply proportionality coefficient. All the sensitivities calculated in this work are in good agreement with those obtained through recalculations.

Projection methods for the numerical solution of Markov chain models

NASA Technical Reports Server (NTRS)

Saad, Youcef

1989-01-01

Projection methods for computing stationary probability distributions for Markov chain models are presented. A general projection method is a method which seeks an approximation from a subspace of small dimension to the original problem. Thus, the original matrix problem of size N is approximated by one of dimension m, typically much smaller than N. A particularly successful class of methods based on this principle is that of Krylov subspace methods which utilize subspaces of the form span(v,av,...,A(exp m-1)v). These methods are effective in solving linear systems and eigenvalue problems (Lanczos, Arnoldi,...) as well as nonlinear equations. They can be combined with more traditional iterative methods such as successive overrelaxation, symmetric successive overrelaxation, or with incomplete factorization methods to enhance convergence.

Spectral simulation of unsteady compressible flow past a circular cylinder

NASA Technical Reports Server (NTRS)

Don, Wai-Sun; Gottlieb, David

1990-01-01

An unsteady compressible viscous wake flow past a circular cylinder was successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for periodic problems is introduced. It was further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It was been shown that the solution of a model problem has exponential growth in time if improper boundary conditions are used. A characteristic boundary condition, which is based on the characteristics of the Euler equations of gas dynamics, was derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data.

Quantum Black Hole Model and HAWKING’S Radiation

NASA Astrophysics Data System (ADS)

Berezin, Victor

The black hole model with a self-gravitating charged spherical symmetric dust thin shell as a source is considered. The Schroedinger-type equation for such a model is derived. This equation appeared to be a finite differences equation. A theory of such an equation is developed and general solution is found and investigated in details. The discrete spectrum of the bound state energy levels is obtained. All the eigenvalues appeared to be infinitely degenerate. The ground state wave functions are evaluated explicitly. The quantum black hole states are selected and investigated. It is shown that the obtained black hole mass spectrum is compatible with the existence of Hawking’s radiation in the limit of low temperatures both for large and nearly extreme Reissner-Nordstrom black holes. The above mentioned infinite degeneracy of the mass (energy) eigenvalues may appeared helpful in resolving the well known information paradox in the black hole physics.

Photonic band gap structure simulator

DOEpatents

Chen, Chiping; Shapiro, Michael A.; Smirnova, Evgenya I.; Temkin, Richard J.; Sirigiri, Jagadishwar R.

2006-10-03

A system and method for designing photonic band gap structures. The system and method provide a user with the capability to produce a model of a two-dimensional array of conductors corresponding to a unit cell. The model involves a linear equation. Boundary conditions representative of conditions at the boundary of the unit cell are applied to a solution of the Helmholtz equation defined for the unit cell. The linear equation can be approximated by a Hermitian matrix. An eigenvalue of the Helmholtz equation is calculated. One computation approach involves calculating finite differences. The model can include a symmetry element, such as a center of inversion, a rotation axis, and a mirror plane. A graphical user interface is provided for the user's convenience. A display is provided to display to a user the calculated eigenvalue, corresponding to a photonic energy level in the Brilloin zone of the unit cell.

A method to stabilize linear systems using eigenvalue gradient information

NASA Technical Reports Server (NTRS)

Wieseman, C. D.

1985-01-01

Formal optimization methods and eigenvalue gradient information are used to develop a stabilizing control law for a closed loop linear system that is initially unstable. The method was originally formulated by using direct, constrained optimization methods with the constraints being the real parts of the eigenvalues. However, because of problems in trying to achieve stabilizing control laws, the problem was reformulated to be solved differently. The method described uses the Davidon-Fletcher-Powell minimization technique to solve an indirect, constrained minimization problem in which the performance index is the Kreisselmeier-Steinhauser function of the real parts of all the eigenvalues. The method is applied successfully to solve two different problems: the determination of a fourth-order control law stabilizes a single-input single-output active flutter suppression system and the determination of a second-order control law for a multi-input multi-output lateral-directional flight control system. Various sets of design variables and initial starting points were chosen to show the robustness of the method.

Effective dimensional reduction algorithm for eigenvalue problems for thin elastic structures: A paradigm in three dimensions

PubMed Central

Ovtchinnikov, Evgueni E.; Xanthis, Leonidas S.

2000-01-01

We present a methodology for the efficient numerical solution of eigenvalue problems of full three-dimensional elasticity for thin elastic structures, such as shells, plates and rods of arbitrary geometry, discretized by the finite element method. Such problems are solved by iterative methods, which, however, are known to suffer from slow convergence or even convergence failure, when the thickness is small. In this paper we show an effective way of resolving this difficulty by invoking a special preconditioning technique associated with the effective dimensional reduction algorithm (EDRA). As an example, we present an algorithm for computing the minimal eigenvalue of a thin elastic plate and we show both theoretically and numerically that it is robust with respect to both the thickness and discretization parameters, i.e. the convergence does not deteriorate with diminishing thickness or mesh refinement. This robustness is sine qua non for the efficient computation of large-scale eigenvalue problems for thin elastic structures. PMID:10655469

Planar dynamics of a uniform beam with rigid bodies affixed to the ends

NASA Technical Reports Server (NTRS)

Storch, J.; Gates, S.

1983-01-01

The planar dynamics of a uniform elastic beam subject to a variety of geometric and natural boundary conditions and external excitations were analyzed. The beams are inextensible and capable of small transverse bending deformations only. Classical beam vibration eigenvalue problems for a cantilever with tip mass, a cantilever with tip body and an unconstrained beam with rigid bodies at each are examined. The characteristic equations, eigenfunctions and orthogonality relations for each are derived. The forced vibration of a cantilever with tip body subject to base acceleration is analyzed. The exact solution of the governing nonhomogeneous partial differential equation with time dependent boundary conditions is presented and compared with a Rayleigh-Ritz approximate solution. The arbitrary planar motion of an elastic beam with rigid bodies at the ends is addressed. Equations of motion are derived for two modal expansions of the beam deflection. The motion equations are cast in a first order form suitable for numerical integration. Selected FORTRAN programs are provided.

Simultaneous multigrid techniques for nonlinear eigenvalue problems: Solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions

NASA Astrophysics Data System (ADS)

Costiner, Sorin; Ta'asan, Shlomo

1995-07-01

Algorithms for nonlinear eigenvalue problems (EP's) often require solving self-consistently a large number of EP's. Convergence difficulties may occur if the solution is not sought in an appropriate region, if global constraints have to be satisfied, or if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP's obtained from discretizations of partial differential EP have often been shown to be more efficient than single level algorithms. This paper presents MG techniques and a MG algorithm for nonlinear Schrödinger Poisson EP's. The algorithm overcomes the above mentioned difficulties combining the following techniques: a MG simultaneous treatment of the eigenvectors and nonlinearity, and with the global constrains; MG stable subspace continuation techniques for the treatment of nonlinearity; and a MG projection coupled with backrotations for separation of solutions. These techniques keep the solutions in an appropriate region, where the algorithm converges fast, and reduce the large number of self-consistent iterations to only a few or one MG simultaneous iteration. The MG projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues. Computational examples for the nonlinear Schrödinger-Poisson EP in two and three dimensions, presenting special computational difficulties that are due to the nonlinearity and to the equal and closely clustered eigenvalues are demonstrated. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N and for the corresponding eigenvalues. One MG simultaneous cycle per fine level was performed. The total computational cost is equivalent to only a few Gauss-Seidel relaxations per eigenvector. An asymptotic convergence rate of 0.15 per MG cycle is attained.

Asymptotics of empirical eigenstructure for high dimensional spiked covariance.

PubMed

Wang, Weichen; Fan, Jianqing

2017-06-01

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies.

Asymptotics of empirical eigenstructure for high dimensional spiked covariance

PubMed Central

Wang, Weichen

2017-01-01

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies. PMID:28835726

On a local solvability and stability of the inverse transmission eigenvalue problem

NASA Astrophysics Data System (ADS)

Bondarenko, Natalia; Buterin, Sergey

2017-11-01

We prove a local solvability and stability of the inverse transmission eigenvalue problem posed by McLaughlin and Polyakov (1994 J. Diff. Equ. 107 351-82). In particular, this result establishes the minimality of the data used therein. The proof is constructive.

A new method of imposing boundary conditions for hyperbolic equations

NASA Technical Reports Server (NTRS)

Funaro, D.; ative.

1987-01-01

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications

NASA Technical Reports Server (NTRS)

Ibrahim, A. H.; Hou, G. J.-W.; Tiwari, S. N. (Principal Investigator)

1996-01-01

Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Sensitivity analysis and approximation methods for general eigenvalue problems

NASA Technical Reports Server (NTRS)

Murthy, D. V.; Haftka, R. T.

1986-01-01

Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought.

Matrix with Prescribed Eigenvectors

ERIC Educational Resources Information Center

Ahmad, Faiz

2011-01-01

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…

Multigrid techniques for nonlinear eigenvalue probems: Solutions of a nonlinear Schroedinger eigenvalue problem in 2D and 3D

NASA Technical Reports Server (NTRS)

Costiner, Sorin; Taasan, Shlomo

1994-01-01

This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.

Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

NASA Technical Reports Server (NTRS)

Antar, B. N.; Fowlis, W. W.

1985-01-01

A three-dimensional, linear stability analysis of a baroclinic flow for Richardson number, Ri, of order unity is presented. The model considered is a thin horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the complete set of governing, nonlinear equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in a closed form. The stability analysis is also based on the complete set of equations; and perturbation possessing zonal, meridional, and vertical structures were considered. Numerical methods were developed for the stability problem which results in a stiff, eighth-order, ordinary differential eigenvalue problem. The previous work on three-dimensional baroclinic instability for small Ri was extended to a more realistic model involving the Prandtl number, sigma, and the Ekman number, E, and to finite growth rates and a wider range of the zonal wavenumber.

NASA Astrophysics Data System (ADS)

2018-05-01

Eigenvalues and eigenvectors, together, constitute the eigenstructure of the system. The design of vibrating systems aimed at satisfying specifications on eigenvalues and eigenvectors, which is commonly known as eigenstructure assignment, has drawn increasing interest over the recent years. The most natural mathematical framework for such problems is constituted by the inverse eigenproblems, which consist in the determination of the system model that features a desired set of eigenvalues and eigenvectors. Although such a problem is intrinsically challenging, several solutions have been proposed in the literature. The approaches to eigenstructure assignment can be basically divided into passive control and active control.

A Projection free method for Generalized Eigenvalue Problem with a nonsmooth Regularizer.

PubMed

Hwang, Seong Jae; Collins, Maxwell D; Ravi, Sathya N; Ithapu, Vamsi K; Adluru, Nagesh; Johnson, Sterling C; Singh, Vikas

2015-12-01

Eigenvalue problems are ubiquitous in computer vision, covering a very broad spectrum of applications ranging from estimation problems in multi-view geometry to image segmentation. Few other linear algebra problems have a more mature set of numerical routines available and many computer vision libraries leverage such tools extensively. However, the ability to call the underlying solver only as a "black box" can often become restrictive. Many 'human in the loop' settings in vision frequently exploit supervision from an expert, to the extent that the user can be considered a subroutine in the overall system. In other cases, there is additional domain knowledge, side or even partial information that one may want to incorporate within the formulation. In general, regularizing a (generalized) eigenvalue problem with such side information remains difficult. Motivated by these needs, this paper presents an optimization scheme to solve generalized eigenvalue problems (GEP) involving a (nonsmooth) regularizer. We start from an alternative formulation of GEP where the feasibility set of the model involves the Stiefel manifold. The core of this paper presents an end to end stochastic optimization scheme for the resultant problem. We show how this general algorithm enables improved statistical analysis of brain imaging data where the regularizer is derived from other 'views' of the disease pathology, involving clinical measurements and other image-derived representations.

Solution of Inverse Kinematics for 6R Robot Manipulators With Offset Wrist Based on Geometric Algebra.

PubMed

Fu, Zhongtao; Yang, Wenyu; Yang, Zhen

2013-08-01

In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. The experimental results show that this method has a large advantage over the classical methods on geometric intuition, computation and real-time, and can be directly extended to all serial robot manipulators and completely automatized, which provides a new tool on the analysis and application of general robot manipulators.

Analytical Solutions of the Schrödinger Equation for the Manning-Rosen plus Hulthén Potential Within SUSY Quantum Mechanics

NASA Astrophysics Data System (ADS)

Ahmadov, A. I.; Naeem, Maria; Qocayeva, M. V.; Tarverdiyeva, V. A.

2018-02-01

In this paper, the bound state solution of the modified radial Schrödinger equation is obtained for the Manning-Rosen plus Hulthén potential by implementing the novel improved scheme to surmount the centrifugal term. The energy eigenvalues and corresponding radial wave functions are defined for any l ≠ 0 angular momentum case via the Nikiforov-Uvarov (NU) and supersymmetric quantum mechanics (SUSYQM) methods. By using these two different methods, equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is demonstrated. The energy levels are worked out and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary l states. A closed form of the normalization constant of the wave functions is also found. It is shown that, the energy eigenvalues and eigenfunctions are sensitive to nr radial and l orbital quantum numbers.

Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials

NASA Astrophysics Data System (ADS)

Yang, Jianke; Nixon, Sean

2016-11-01

Stability of soliton families in one-dimensional nonlinear Schrödinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically. It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition. In addition, a pseudo-Hamiltonian-Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian-Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian. The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets (λ , - λ ,λ* , -λ*), similar to conservative systems and PT-symmetric systems. This quartet eigenvalue symmetry is very surprising for non- PT-symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.

Asymptotic Behaviour of the Ground State of Singularly Perturbed Elliptic Equations

NASA Astrophysics Data System (ADS)

Piatnitski, Andrey L.

The ground state of a singularly perturbed nonselfadjoint elliptic operator defined on a smooth compact Riemannian manifold with metric aij(x)=(aij(x))-1, is studied. We investigate the limiting behaviour of the first eigenvalue of this operator as μ goes to zero, and find the logarithmic asymptotics of the first eigenfunction everywhere on the manifold. The results are formulated in terms of auxiliary variational problems on the manifold. This approach also allows to study the general singularly perturbed second order elliptic operator on a bounded domain in Rn.

Aircraft model prototypes which have specified handling-quality time histories

NASA Technical Reports Server (NTRS)

Johnson, S. H.

1978-01-01

Several techniques for obtaining linear constant-coefficient airplane models from specified handling-quality time histories are discussed. The pseudodata method solves the basic problem, yields specified eigenvalues, and accommodates state-variable transfer-function zero suppression. The algebraic equations to be solved are bilinear, at worst. The disadvantages are reduced generality and no assurance that the resulting model will be airplane like in detail. The method is fully illustrated for a fourth-order stability-axis small motion model with three lateral handling quality time histories specified. The FORTRAN program which obtains and verifies the model is included and fully documented.

A useful observable for estimating keff in fast subcritical systems

NASA Astrophysics Data System (ADS)

Saracco, Paolo; Borreani, Walter; Chersola, Davide; Lomonaco, Guglielmo; Ricco, Gianni; Ripani, Marco

2017-09-01

The neutron multiplication factor keff is a key quantity to characterize subcritical neutron multiplying devices and for understanting their physical behaviour, being related to the fundamental eigenvalue of Boltzmann transport equation. Both the maximum available power - and all quantities related to it, like, e.g. the effectiveness in burning nuclear wastes - as well as reactor kinetics and dynamics depend on keff. Nevertheless, keff is not directly measurable and its determination results from the solution of an inverse problem: minimizing model dependence of the solution for keff then becomes a critical issue, relevant both for practical and theoretical reasons.

Nonlinear Equations of Motion for Cantilever Rotor Blades in Hover with Pitch Link Flexibility, Twist, Precone, Droop, Sweep, Torque Offset, and Blade Root Offset

NASA Technical Reports Server (NTRS)

Hodges, D. H.

1976-01-01

Nonlinear equations of motion for a cantilever rotor blade are derived for the hovering flight condition. The blade is assumed to have twist, precone, droop, sweep, torque offset and blade root offset, and the elastic axis and the axes of center of mass, tension, and aerodynamic center coincident at the quarter chord. The blade is cantilevered in bending, but has a torsional root spring to simulate pitch link flexibility. Aerodynamic forces acting on the blade are derived from strip theory based on quasi-steady two-dimensional airfoil theory. The equations are hybrid, consisting of one integro-differential equation for root torsion and three integro-partial differential equations for flatwise and chordwise bending and elastic torsion. The equations are specialized for a uniform blade and reduced to nonlinear ordinary differential equations by Galerkin's method. They are linearized for small perturbation motions about the equilibrium operating condition. Modal analysis leads to formulation of a standard eigenvalue problem where the elements of the stability matrix depend on the solution of the equilibrium equations. Two different forms of the root torsion equation are derived that yield virtually identical numerical results. This provides a reasonable check for the accuracy of the equations.

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

NASA Astrophysics Data System (ADS)

Zhang, Linghai

2017-10-01

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 < 2 (1 + αγ) θ < αβγ; the existence and stability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and γ2 ε > 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 <γ2 ε < 1; the existence and instability of an upside down standing pulse solution if 0 < (1 + αγ) θ < αβγ < 2 (1 + αγ) θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 < 2 θ < β; the existence and stability of two standing wave fronts if 2 θ = β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 < θ < β < 2 θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ] -w0, where w0 = α (β - 2 θ) > 0 is a positive constant, if 0 < 2 θ < β. To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear differential operator, if and only if λ0 is a zero of the Evans function. The stability, instability and bifurcations of the nonlinear waves follow from the zeros of the Evans functions. A very important motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.

Generalized vector calculus on convex domain

NASA Astrophysics Data System (ADS)

Agrawal, Om P.; Xu, Yufeng

2015-06-01

In this paper, we apply recently proposed generalized integral and differential operators to develop generalized vector calculus and generalized variational calculus for problems defined over a convex domain. In particular, we present some generalization of Green's and Gauss divergence theorems involving some new operators, and apply these theorems to generalized variational calculus. For fractional power kernels, the formulation leads to fractional vector calculus and fractional variational calculus for problems defined over a convex domain. In special cases, when certain parameters take integer values, we obtain formulations for integer order problems. Two examples are presented to demonstrate applications of the generalized variational calculus which utilize the generalized vector calculus developed in the paper. The first example leads to a generalized partial differential equation and the second example leads to a generalized eigenvalue problem, both in two dimensional convex domains. We solve the generalized partial differential equation by using polynomial approximation. A special case of the second example is a generalized isoperimetric problem. We find an approximate solution to this problem. Many physical problems containing integer order integrals and derivatives are defined over arbitrary domains. We speculate that future problems containing fractional and generalized integrals and derivatives in fractional mechanics will be defined over arbitrary domains, and therefore, a general variational calculus incorporating a general vector calculus will be needed for these problems. This research is our first attempt in that direction.

Eigenvalue approach to coupled thermoelasticity in a rotating isotropic medium

NASA Astrophysics Data System (ADS)

Bayones, F. S.; Abd-Alla, A. M.

2018-03-01

In this paper the linear theory of the thermoelasticity has been employed to study the effect of the rotation in a thermoelastic half-space containing heat source on the boundary of the half-space. It is assumed that the medium under consideration is traction free, homogeneous, isotropic, as well as without energy dissipation. The normal mode analysis has been applied in the basic equations of coupled thermoelasticity and finally the resulting equations are written in the form of a vector- matrix differential equation which is then solved by eigenvalue approach. Numerical results for the displacement components, stresses, and temperature are given and illustrated graphically. Comparison was made with the results obtained in the presence and absence of the rotation. The results indicate that the effect of rotation, non-dimensional thermal wave and time are very pronounced.

Nonunitary and unitary approach to Eigenvalue problem of Boson operators and squeezed coherent states

NASA Technical Reports Server (NTRS)

Wunsche, A.

1993-01-01

The eigenvalue problem of the operator a + zeta(boson creation operator) is solved for arbitrarily complex zeta by applying a nonunitary operator to the vacuum state. This nonunitary approach is compared with the unitary approach leading for the absolute value of zeta less than 1 to squeezed coherent states.

Robustness of linear quadratic state feedback designs in the presence of system uncertainty. [application to Augmentor Wing Jet STOL Research Aircraft flare control autopilot design

NASA Technical Reports Server (NTRS)

Patel, R. V.; Toda, M.; Sridhar, B.

1977-01-01

The paper deals with the problem of expressing the robustness (stability) property of a linear quadratic state feedback (LQSF) design quantitatively in terms of bounds on the perturbations (modeling errors or parameter variations) in the system matrices so that the closed-loop system remains stable. Nonlinear time-varying and linear time-invariant perturbations are considered. The only computation required in obtaining a measure of the robustness of an LQSF design is to determine the eigenvalues of two symmetric matrices determined when solving the algebraic Riccati equation corresponding to the LQSF design problem. Results are applied to a complex dynamic system consisting of the flare control of a STOL aircraft. The design of the flare control is formulated as an LQSF tracking problem.

Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Swanson, R. C.; Turkel, Eli

1987-01-01

An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined. Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flow is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.

Large space structure damping design

NASA Technical Reports Server (NTRS)

Pilkey, W. D.; Haviland, J. K.

1983-01-01

Several FORTRAN subroutines and programs were developed which compute complex eigenvalues of a damped system using different approaches, and which rescale mode shapes to unit generalized mass and make rigid bodies orthogonal to each other. An analytical proof of a Minimum Constrained Frequency Criterion (MCFC) for a single damper is presented. A method to minimize the effect of control spill-over for large space structures is proposed. The characteristic equation of an undamped system with a generalized control law is derived using reanalysis theory. This equation can be implemented in computer programs for efficient eigenvalue analysis or control quasi synthesis. Methods to control vibrations in large space structure are reviewed and analyzed. The resulting prototype, using electromagnetic actuator, is described.

Modal interaction in linear dynamic systems near degenerate modes

NASA Technical Reports Server (NTRS)

Afolabi, D.

1991-01-01

In various problems in structural dynamics, the eigenvalues of a linear system depend on a characteristic parameter of the system. Under certain conditions, two eigenvalues of the system approach each other as the characteristic parameter is varied, leading to modal interaction. In a system with conservative coupling, the two eigenvalues eventually repel each other, leading to the curve veering effect. In a system with nonconservative coupling, the eigenvalues continue to attract each other, eventually colliding, leading to eigenvalue degeneracy. Modal interaction is studied in linear systems with conservative and nonconservative coupling using singularity theory, sometimes known as catastrophe theory. The main result is this: eigenvalue degeneracy is a cause of instability; in systems with conservative coupling, it induces only geometric instability, whereas in systems with nonconservative coupling, eigenvalue degeneracy induces both geometric and elastic instability. Illustrative examples of mechanical systems are given.

Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation

NASA Astrophysics Data System (ADS)

Benner, Peter; Dolgov, Sergey; Khoromskaia, Venera; Khoromskij, Boris N.

2017-04-01

In this paper, we propose and study two approaches to approximate the solution of the Bethe-Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe-Salpeter operator. The complexity is reduced to O (Nb2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O (Nb6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, No

Multitasking the Davidson algorithm for the large, sparse eigenvalue problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Umar, V.M.; Fischer, C.F.

1989-01-01

The authors report how the Davidson algorithm, developed for handling the eigenvalue problem for large and sparse matrices arising in quantum chemistry, was modified for use in atomic structure calculations. To date these calculations have used traditional eigenvalue methods, which limit the range of feasible calculations because of their excessive memory requirements and unsatisfactory performance attributed to time-consuming and costly processing of zero valued elements. The replacement of a traditional matrix eigenvalue method by the Davidson algorithm reduced these limitations. Significant speedup was found, which varied with the size of the underlying problem and its sparsity. Furthermore, the range ofmore » matrix sizes that can be manipulated efficiently was expended by more than one order or magnitude. On the CRAY X-MP the code was vectorized and the importance of gather/scatter analyzed. A parallelized version of the algorithm obtained an additional 35% reduction in execution time. Speedup due to vectorization and concurrency was also measured on the Alliant FX/8.« less

An analysis of the vertical structure equation for arbitrary thermal profiles

NASA Technical Reports Server (NTRS)

Cohn, Stephen E.; Dee, Dick P.

1989-01-01

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.

An analysis of the vertical structure equation for arbitrary thermal profiles

NASA Technical Reports Server (NTRS)

Cohn, Stephen E.; Dee, Dick P.

1987-01-01

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.

A study of the response of nonlinear springs

NASA Technical Reports Server (NTRS)

Hyer, M. W.; Knott, T. W.; Johnson, E. R.

1991-01-01

The various phases to developing a methodology for studying the response of a spring-reinforced arch subjected to a point load are discussed. The arch is simply supported at its ends with both the spring and the point load assumed to be at midspan. The spring is present to off-set the typical snap through behavior normally associated with arches, and to provide a structure that responds with constant resistance over a finite displacement. The various phases discussed consist of the following: (1) development of the closed-form solution for the shallow arch case; (2) development of a finite difference analysis to study (shallow) arches; and (3) development of a finite element analysis for studying more general shallow and nonshallow arches. The two numerical analyses rely on a continuation scheme to move the solution past limit points, and to move onto bifurcated paths, both characteristics being common to the arch problem. An eigenvalue method is used for a continuation scheme. The finite difference analysis is based on a mixed formulation (force and displacement variables) of the governing equations. The governing equations for the mixed formulation are in first order form, making the finite difference implementation convenient. However, the mixed formulation is not well-suited for the eigenvalue continuation scheme. This provided the motivation for the displacement based finite element analysis. Both the finite difference and the finite element analyses are compared with the closed form shallow arch solution. Agreement is excellent, except for the potential problems with the finite difference analysis and the continuation scheme. Agreement between the finite element analysis and another investigator's numerical analysis for deep arches is also good.

Linear Stability Analysis of Gravitational Effects on a Low-Density Gas Jet Injected into a High-Density Medium

NASA Technical Reports Server (NTRS)

Lawson, Anthony L.; Parthasarathy, Ramkumar N.

2005-01-01

The objective of this study was to determine the effects of buoyancy on the absolute instability of low-density gas jets injected into high-density gas mediums. Most of the existing analyses of low-density gas jets injected into a high-density ambient have been carried out neglecting effects of gravity. In order to investigate the influence of gravity on the near-injector development of the flow, a spatio-temporal stability analysis of a low-density round jet injected into a high-density ambient gas was performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The Briggs-Bers criterion was combined with the spatio-temporal stability analysis to determine the nature of the absolute instability of the jet whether absolutely or convectively unstable. The roles of the density ratio, Froude number, Schmidt number, and the lateral shift between the density and velocity profiles on the absolute instability of the jet were determined. Comparisons of the results with previous experimental studies show good agreement when the effects of these variables are combined together. Thus, the combination of these variables determines how absolutely unstable the jet will be.

Inflationary dynamics for matrix eigenvalue problems

PubMed Central

Heller, Eric J.; Kaplan, Lev; Pollmann, Frank

2008-01-01

Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the acoustic modes of a concert hall, or hundreds of other physical quantities. Often only the few eigenpairs with the lowest or highest frequency (extremal solutions) are needed. Methods that have been developed over the past 60 years to solve such problems include the Lanczos algorithm, Jacobi–Davidson techniques, and the conjugate gradient method. Here, we present a way to solve the extremal eigenvalue/eigenvector problem, turning it into a nonlinear classical mechanical system with a modified Lagrangian constraint. The constraint induces exponential inflationary growth of the desired extremal solutions. PMID:18511564

Free Vibration Study of Anti-Symmetric Angle-Ply Laminated Plates under Clamped Boundary Conditions

NASA Astrophysics Data System (ADS)

Viswanathan, K. K.; Karthik, K.; Sanyasiraju, Y. V. S. S.; Aziz, Z. A.

2016-11-01

Two type of numerical approach namely, Radial Basis Function and Spline approximation, used to analyse the free vibration of anti-symmetric angle-ply laminated plates under clamped boundary conditions. The equations of motion are derived using YNS theory under first order shear deformation. By assuming the solution in separable form, coupled differential equations obtained in term of mid-plane displacement and rotational functions. The coupled differential is then approximated using Spline function and radial basis function to obtain the generalize eigenvalue problem and parametric studies are made to investigate the effect of aspect ratio, length-to-thickness ratio, number of layers, fibre orientation and material properties with respect to the frequency parameter. Some results are compared with the existing literature and other new results are given in tables and graphs.

Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method.

PubMed

Guizal, Brahim; Barchiesi, Dominique; Felbacq, Didier

2003-12-01

We have developed a new formulation of the coupled-wave method (CWM) to handle aperiodic lamellar structures, and it will be referred to as the aperiodic coupled-wave method (ACWM). The space is still divided into three regions, but the fields are written by use of their Fourier integrals instead of the Fourier series. In the modulated region the relative permittivity is represented by its Fourier transform, and then a set of integro-differential equations is derived. Discretizing the last system leads to a set of ordinary differential equations that is reduced to an eigenvalue problem, as is usually done in the CWM. To assess the method, we compare our results with three independent formalisms: the Rayleigh perturbation method for small samples, the volume integral method, and the finite-element method.

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra by quantum separation of variables

NASA Astrophysics Data System (ADS)

Pezelier, Baptiste

2018-02-01

In this proceeding, we recall the notion of quantum integrable systems on a lattice and then introduce the Sklyanin’s Separation of Variables method. We sum up the main results for the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. These results apply as well to the spectral analysis of the lattice sine-Gordon model with open boundary conditions. The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations. We state an equivalent characterization as the set of solutions to a Baxter’s like T-Q functional equation, allowing us to rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form.

nu-TRLan User Guide Version 1.0: A High-Performance Software Package for Large-Scale Harmitian Eigenvalue Problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yamazaki, Ichitaro; Wu, Kesheng; Simon, Horst

2008-10-27

The original software package TRLan, [TRLan User Guide], page 24, implements the thick restart Lanczos method, [Wu and Simon 2001], page 24, for computing eigenvalues {lambda} and their corresponding eigenvectors v of a symmetric matrix A: Av = {lambda}v. Its effectiveness in computing the exterior eigenvalues of a large matrix has been demonstrated, [LBNL-42982], page 24. However, its performance strongly depends on the user-specified dimension of a projection subspace. If the dimension is too small, TRLan suffers from slow convergence. If it is too large, the computational and memory costs become expensive. Therefore, to balance the solution convergence and costs,more » users must select an appropriate subspace dimension for each eigenvalue problem at hand. To free users from this difficult task, nu-TRLan, [LNBL-1059E], page 23, adjusts the subspace dimension at every restart such that optimal performance in solving the eigenvalue problem is automatically obtained. This document provides a user guide to the nu-TRLan software package. The original TRLan software package was implemented in Fortran 90 to solve symmetric eigenvalue problems using static projection subspace dimensions. nu-TRLan was developed in C and extended to solve Hermitian eigenvalue problems. It can be invoked using either a static or an adaptive subspace dimension. In order to simplify its use for TRLan users, nu-TRLan has interfaces and features similar to those of TRLan: (1) Solver parameters are stored in a single data structure called trl-info, Chapter 4 [trl-info structure], page 7. (2) Most of the numerical computations are performed by BLAS, [BLAS], page 23, and LAPACK, [LAPACK], page 23, subroutines, which allow nu-TRLan to achieve optimized performance across a wide range of platforms. (3) To solve eigenvalue problems on distributed memory systems, the message passing interface (MPI), [MPI forum], page 23, is used. The rest of this document is organized as follows. In Chapter 2 [Installation], page 2, we provide an installation guide of the nu-TRLan software package. In Chapter 3 [Example], page 3, we present a simple nu-TRLan example program. In Chapter 4 [trl-info structure], page 7, and Chapter 5 [trlan subroutine], page 14, we describe the solver parameters and interfaces in detail. In Chapter 6 [Solver parameters], page 21, we discuss the selection of the user-specified parameters. In Chapter 7 [Contact information], page 22, we give the acknowledgements and contact information of the authors. In Chapter 8 [References], page 23, we list reference to related works.« less

Sensitivity of coronal loop sausage mode frequencies and decay rates to radial and longitudinal density inhomogeneities: a spectral approach

NASA Astrophysics Data System (ADS)

Cally, Paul S.; Xiong, Ming

2018-01-01

Fast sausage modes in solar magnetic coronal loops are only fully contained in unrealistically short dense loops. Otherwise they are leaky, losing energy to their surrounds as outgoing waves. This causes any oscillation to decay exponentially in time. Simultaneous observations of both period and decay rate therefore reveal the eigenfrequency of the observed mode, and potentially insight into the tubes’ nonuniform internal structure. In this article, a global spectral description of the oscillations is presented that results in an implicit matrix eigenvalue equation where the eigenvalues are associated predominantly with the diagonal terms of the matrix. The off-diagonal terms vanish identically if the tube is uniform. A linearized perturbation approach, applied with respect to a uniform reference model, is developed that makes the eigenvalues explicit. The implicit eigenvalue problem is easily solved numerically though, and it is shown that knowledge of the real and imaginary parts of the eigenfrequency is sufficient to determine the width and density contrast of a boundary layer over which the tubes’ enhanced internal densities drop to ambient values. Linearized density kernels are developed that show sensitivity only to the extreme outside of the loops for radial fundamental modes, especially for small density enhancements, with no sensitivity to the core. Higher radial harmonics do show some internal sensitivity, but these will be more difficult to observe. Only kink modes are sensitive to the tube centres. Variation in internal and external Alfvén speed along the loop is shown to have little effect on the fundamental dimensionless eigenfrequency, though the associated eigenfunction becomes more compact at the loop apex as stratification increases, or may even displace from the apex.

The generalized DMPK equation revisited: towards a systematic derivation

NASA Astrophysics Data System (ADS)

Douglas, Andrew; Markoš, Peter; Muttalib, K. A.

2014-03-01

The generalized Dorokov-Mello-Pereyra-Kumar (DMPK) equation has recently been used to obtain a family of very broad and highly asymmetric conductance distributions for three-dimensional disordered conductors. However, there are two major criticisms of the derivation of the generalized DMPK equation: (1) certain eigenvector correlations were neglected based on qualitative arguments that cannot be valid for all strengths of disorder, and (2) the repulsion between two closely spaced eigenvalues were not rigorously governed by symmetry considerations. In this work we show that it is possible to address both criticisms by including the eigenvalue and eigenvector correlations in a systematic and controlled way. It turns out that the added correlations determine the evolution of the Jacobian, without affecting the evaluation of the conductance distributions. They also guarantee the symmetry requirements. In addition, we obtain an exact relationship between the eigenvectors and the Lyapunov exponents leading to a sum rule for the latter at all disorder strengths.

The Theory of Quantized Fields. III

DOE R&D Accomplishments Database

Schwinger, J.

1953-05-01

In this paper we discuss the electromagnetic field, as perturbed by a prescribed current. All quantities of physical interest in various situations, eigenvalues, eigenfunctions, and transformation probabilities, are derived from a general transformation function which is expressed in a non-Hermitian representation. The problems treated are: the determination of the energy-momentum eigenvalues and eigenfunctions for the isolated electromagnetic field, and the energy eigenvalues and eigenfunctions for the field perturbed by a time-independent current that departs from zero only within a finite time interval, and for a time-dependent current that assumes non-vanishing time-independent values initially and finally. The results are applied in a discussion of the intra-red catastrophe and of the adiabatic theorem. It is shown how the latter can be exploited to give a uniform formulation for all problems requiring the evaluation of transition probabilities or eigenvalue displacements.

Biological Applications in the Mathematics Curriculum

ERIC Educational Resources Information Center

Marland, Eric; Palmer, Katrina M.; Salinas, Rene A.

2008-01-01

In this article we provide two detailed examples of how we incorporate biological examples into two mathematics courses: Linear Algebra and Ordinary Differential Equations. We use Leslie matrix models to demonstrate the biological properties of eigenvalues and eigenvectors. For Ordinary Differential Equations, we show how using a logistic growth…

Combined Effects Aluminized Explosives

DTIC Science & Technology

2010-07-01

1 4 5 AREA EXPANSIONS Figure 4 Cylinder velocities for PAX-3 (left) and an empirical PAX-30 JWL (right) THERMODYNAMIC EQUATIONS OF...STATE The JWLB and Jones-Wilkins-Lee ( JWL ) equations of state were parameterized for combined effects explosives using fairly conventional methodology...state. Such warning messages should be ignored when using these JWLB and JWL equations of state representing eigenvalue detonation behavior. Table 1

Analysis of physics-based preconditioning for single-phase subchannel equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hansel, J. E.; Ragusa, J. C.; Allu, S.

2013-07-01

The (single-phase) subchannel approximations are used throughout nuclear engineering to provide an efficient flow simulation because the computational burden is much smaller than for computational fluid dynamics (CFD) simulations, and empirical relations have been developed and validated to provide accurate solutions in appropriate flow regimes. Here, the subchannel equations have been recast in a residual form suitable for a multi-physics framework. The Eigen spectrum of the Jacobian matrix, along with several potential physics-based preconditioning approaches, are evaluated, and the the potential for improved convergence from preconditioning is assessed. The physics-based preconditioner options include several forms of reduced equations that decouplemore » the subchannels by neglecting crossflow, conduction, and/or both turbulent momentum and energy exchange between subchannels. Eigen-scopy analysis shows that preconditioning moves clusters of eigenvalues away from zero and toward one. A test problem is run with and without preconditioning. Without preconditioning, the solution failed to converge using GMRES, but application of any of the preconditioners allowed the solution to converge. (authors)« less

Stability of a laminar premixed supersonic free shear layer with chemical reactions

NASA Technical Reports Server (NTRS)

Menon, S.; Anderson, J. D., Jr.; Pai, S. I.

1984-01-01

The stability of a two-dimensional compressible supersonic flow in the wake of a flat plate is discussed. The fluid is a multi-species mixture which is undergoing finite rate chemical reactions. The spatial stability of an infinitesimal disturbance in the fluid is considered. Numerical solutions of the eigenvalue stability equations for both reactive and nonreactive supersonic flows are presented and discussed. The chemical reactions have significant influence on the stability behavior. For instance, a neutral eigenvalue is observed near the freestream Mach number of 2.375 for the nonreactive case, but disappears when the reaction is turned on. For reactive flows, the eigenvalues are not very dependent on the free stream Mach number.

Analytical and experimental study of the vibration of bonded beams with a lap joint

NASA Technical Reports Server (NTRS)

Rao, M. D.; Crocker, M. J.

1990-01-01

A theoretical model to study the flexural vibration of a bonded lap joint system is described in this paper. First, equations of motion at the joint region are derived using a differential element approach. The transverse displacements of the upper and lower beam are considered to be different. The adhesive is assumed to be linearly viscoelastic and the widely used Kelvin-Voight model is used to represent the viscoelastic behavior of the adhesive. The shear force at the interface between the adhesive and the beam is obtained from the simple bending motion equations of the two beams. The resulting equations of motion are combined with the equations of transverse vibration of the beams in the unjointed regions. These are later solved as a boundary value problem to obtain the eigenvalues and eigenvectors of the system. The model can be used to predict the natural frequencies, modal damping ratios, and mode shapes of the system for free vibration. Good agreement between numerical and experimental results was obtained for a system of graphite epoxy beams lap-jointed by an epoxy adhesive.

Piecewise linear approximation for hereditary control problems

NASA Technical Reports Server (NTRS)

Propst, Georg

1987-01-01

Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

Implicity restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations

NASA Technical Reports Server (NTRS)

Sorensen, Danny C.

1996-01-01

Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

A finite element algorithm for high-lying eigenvalues with Neumann and Dirichlet boundary conditions

NASA Astrophysics Data System (ADS)

Báez, G.; Méndez-Sánchez, R. A.; Leyvraz, F.; Seligman, T. H.

2014-01-01

We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions, or combinations of either for different parts of the boundary. We use an inverse power plus Gauss-Seidel algorithm to solve the generalized eigenvalue problem. For Neumann boundary conditions the method is much more efficient than the equivalent finite difference algorithm. We checked the algorithm by comparing the cumulative level density of the spectrum obtained numerically with the theoretical prediction given by the Weyl formula. We found a systematic deviation due to the discretization, not to the algorithm itself.

A Well-Tempered Hybrid Method for Solving Challenging Time-Dependent Density Functional Theory (TDDFT) Systems.

PubMed

Kasper, Joseph M; Williams-Young, David B; Vecharynski, Eugene; Yang, Chao; Li, Xiaosong

2018-04-10

The time-dependent Hartree-Fock (TDHF) and time-dependent density functional theory (TDDFT) equations allow one to probe electronic resonances of a system quickly and inexpensively. However, the iterative solution of the eigenvalue problem can be challenging or impossible to converge, using standard methods such as the Davidson algorithm for spectrally dense regions in the interior of the spectrum, as are common in X-ray absorption spectroscopy (XAS). More robust solvers, such as the generalized preconditioned locally harmonic residual (GPLHR) method, can alleviate this problem, but at the expense of higher average computational cost. A hybrid method is proposed which adapts to the problem in order to maximize computational performance while providing the superior convergence of GPLHR. In addition, a modification to the GPLHR algorithm is proposed to adaptively choose the shift parameter to enforce a convergence of states above a predefined energy threshold.

A Density Perturbation Method to Study the Eigenstructure of Two-Phase Flow Equation Systems

NASA Astrophysics Data System (ADS)

Cortes, J.; Debussche, A.; Toumi, I.

1998-12-01

Many interesting and challenging physical mechanisms are concerned with the mathematical notion of eigenstructure. In two-fluid models, complex phasic interactions yield a complex eigenstructure which may raise numerous problems in numerical simulations. In this paper, we develop a perturbation method to examine the eigenvalues and eigenvectors of two-fluid models. This original method, based on the stiffness of the density ratio, provides a convenient tool to study the relevance of pressure momentum interactions and allows us to get precise approximations of the whole flow eigendecomposition for minor requirements. Roe scheme is successfully implemented and some numerical tests are presented.

Numerical methods for systems of conservation laws of mixed type using flux splitting

NASA Technical Reports Server (NTRS)

Shu, Chi-Wang

1990-01-01

The essentially non-oscillatory (ENO) finite difference scheme is applied to systems of conservation laws of mixed hyperbolic-elliptic type. A flux splitting, with the corresponding Jacobi matrices having real and positive/negative eigenvalues, is used. The hyperbolic ENO operator is applied separately. The scheme is numerically tested on the van der Waals equation in fluid dynamics. Convergence was observed with good resolution to weak solutions for various Riemann problems, which are then numerically checked to be admissible as the viscosity-capillarity limits. The interesting phenomena of the shrinking of elliptic regions if they are present in the initial conditions were also observed.

Travelling fronts of the CO oxidation on Pd(111) with coverage-dependent diffusion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cisternas, Jaime, E-mail: jecisternas@miuandes.cl; Karpitschka, Stefan; Wehner, Stefan

2014-10-28

In this work, we study a surface reaction on Pd(111) crystals under ultra-high-vacuum conditions that can be modeled by two coupled reaction-diffusion equations. In the bistable regime, the reaction exhibits travelling fronts that can be observed experimentally using photo electron emission microscopy. The spatial profile of the fronts reveals a coverage-dependent diffusivity for one of the species. We propose a method to solve the nonlinear eigenvalue problem and compute the direction and the speed of the fronts based on a geometrical construction in phase-space. This method successfully captures the dependence of the speed on control parameters and diffusivities.

Numerical Solutions of One Reduced Bethe-Salpeter Equation for the Coulombic Bound States Composed of Virtual Constituents

NASA Astrophysics Data System (ADS)

Chen, Jiao-Kai

2018-04-01

We present one reduction of the Bethe-Salpeter equation for the bound states composed of two off-mass-shell constituents. Both the relativistic effects and the virtuality effects can be considered in the obtained spinless virtuality distribution equation. The eigenvalues of the spinless virtuality distribution equation are perturbatively calculated and the bound states e+e-, μ+μ-, τ+τ-, μ+e-, and τ+e- are discussed.

A new approach to the method of source-sink potentials for molecular conduction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pickup, Barry T., E-mail: B.T.Pickup@sheffield.ac.uk, E-mail: P.W.Fowler@sheffield.ac.uk; Fowler, Patrick W., E-mail: B.T.Pickup@sheffield.ac.uk, E-mail: P.W.Fowler@sheffield.ac.uk; Borg, Martha

2015-11-21

We re-derive the tight-binding source-sink potential (SSP) equations for ballistic conduction through conjugated molecular structures in a form that avoids singularities. This enables derivation of new results for families of molecular devices in terms of eigenvectors and eigenvalues of the adjacency matrix of the molecular graph. In particular, we define the transmission of electrons through individual molecular orbitals (MO) and through MO shells. We make explicit the behaviour of the total current and individual MO and shell currents at molecular eigenvalues. A rich variety of behaviour is found. A SSP device has specific insulation or conduction at an eigenvalue ofmore » the molecular graph (a root of the characteristic polynomial) according to the multiplicities of that value in the spectra of four defined device polynomials. Conduction near eigenvalues is dominated by the transmission curves of nearby shells. A shell may be inert or active. An inert shell does not conduct at any energy, not even at its own eigenvalue. Conduction may occur at the eigenvalue of an inert shell, but is then carried entirely by other shells. If a shell is active, it carries all conduction at its own eigenvalue. For bipartite molecular graphs (alternant molecules), orbital conduction properties are governed by a pairing theorem. Inertness of shells for families such as chains and rings is predicted by selection rules based on node counting and degeneracy.« less

Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method

NASA Technical Reports Server (NTRS)

Tischler, V. A.; Venkayya, Vipperla B.

1993-01-01

The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.

On Bi-Grid Local Mode Analysis of Solution Techniques for 3-D Euler and Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Ibraheem, S. O.; Demuren, A. O.

1994-01-01

A procedure is presented for utilizing a bi-grid stability analysis as a practical tool for predicting multigrid performance in a range of numerical methods for solving Euler and Navier-Stokes equations. Model problems based on the convection, diffusion and Burger's equation are used to illustrate the superiority of the bi-grid analysis as a predictive tool for multigrid performance in comparison to the smoothing factor derived from conventional von Neumann analysis. For the Euler equations, bi-grid analysis is presented for three upwind difference based factorizations, namely Spatial, Eigenvalue and Combination splits, and two central difference based factorizations, namely LU and ADI methods. In the former, both the Steger-Warming and van Leer flux-vector splitting methods are considered. For the Navier-Stokes equations, only the Beam-Warming (ADI) central difference scheme is considered. In each case, estimates of multigrid convergence rates from the bi-grid analysis are compared to smoothing factors obtained from single-grid stability analysis. Effects of grid aspect ratio and flow skewness are examined. Both predictions are compared with practical multigrid convergence rates for 2-D Euler and Navier-Stokes solutions based on the Beam-Warming central scheme.

Resolvent analysis of shear flows using One-Way Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Rigas, Georgios; Schmidt, Oliver; Towne, Aaron; Colonius, Tim

2017-11-01

For three-dimensional flows, questions of stability, receptivity, secondary flows, and coherent structures require the solution of large partial-derivative eigenvalue problems. Reduced-order approximations are thus required for engineering prediction since these problems are often computationally intractable or prohibitively expensive. For spatially slowly evolving flows, such as jets and boundary layers, the One-Way Navier-Stokes (OWNS) equations permit a fast spatial marching procedure that results in a huge reduction in computational cost. Here, an adjoint-based optimization framework is proposed and demonstrated for calculating optimal boundary conditions and optimal volumetric forcing. The corresponding optimal response modes are validated against modes obtained in terms of global resolvent analysis. For laminar base flows, the optimal modes reveal modal and non-modal transition mechanisms. For turbulent base flows, they predict the evolution of coherent structures in a statistical sense. Results from the application of the method to three-dimensional laminar wall-bounded flows and turbulent jets will be presented. This research was supported by the Office of Naval Research (N00014-16-1-2445) and Boeing Company (CT-BA-GTA-1).

A problem with inverse time for a singularly perturbed integro-differential equation with diagonal degeneration of the kernel of high order

NASA Astrophysics Data System (ADS)

Bobodzhanov, A. A.; Safonov, V. F.

2016-04-01

We consider an algorithm for constructing asymptotic solutions regularized in the sense of Lomov (see [1], [2]). We show that such problems can be reduced to integro-differential equations with inverse time. But in contrast to known papers devoted to this topic (see, for example, [3]), in this paper we study a fundamentally new case, which is characterized by the absence, in the differential part, of a linear operator that isolates, in the asymptotics of the solution, constituents described by boundary functions and by the fact that the integral operator has kernel with diagonal degeneration of high order. Furthermore, the spectrum of the regularization operator A(t) (see below) may contain purely imaginary eigenvalues, which causes difficulties in the application of the methods of construction of asymptotic solutions proposed in the monograph [3]. Based on an analysis of the principal term of the asymptotics, we isolate a class of inhomogeneities and initial data for which the exact solution of the original problem tends to the limit solution (as \\varepsilon\\to+0) on the entire time interval under consideration, also including a boundary-layer zone (that is, we solve the so-called initialization problem). The paper is of a theoretical nature and is designed to lead to a greater understanding of the problems in the theory of singular perturbations. There may be applications in various applied areas where models described by integro-differential equations are used (for example, in elasticity theory, the theory of electrical circuits, and so on).

Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems

NASA Astrophysics Data System (ADS)

Bäcker, A.

Summary: We give an introduction to some of the numerical aspects in quantum chaos. The classical dynamics of two-dimensional area-preserving maps on the torus is illustrated using the standard map and a perturbed cat map. The quantization of area-preserving maps given by their generating function is discussed and for the computation of the eigenvalues a computer program in Python is presented. We illustrate the eigenvalue distribution for two types of perturbed cat maps, one leading to COE and the other to CUE statistics. For the eigenfunctions of quantum maps we study the distribution of the eigenvectors and compare them with the corresponding random matrix distributions. The Husimi representation allows for a direct comparison of the localization of the eigenstates in phase space with the corresponding classical structures. Examples for a perturbed cat map and the standard map with different parameters are shown. Billiard systems and the corresponding quantum billiards are another important class of systems (which are also relevant to applications, for example in mesoscopic physics). We provide a detailed exposition of the boundary integral method, which is one important method to determine the eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several methods to determine the eigenvalues from the Fredholm equation and illustrate them for the stadium billiard. The occurrence of spurious solutions is discussed in detail and illustrated for the circular billiard, the stadium billiard, and the annular sector billiard. We emphasize the role of the normal derivative function to compute the normalization of eigenfunctions, momentum representations or autocorrelation functions in a very efficient and direct way. Some examples for these quantities are given and discussed.

Boundary-value problems in ODE

NASA Astrophysics Data System (ADS)

Tanriverdi, Tanfer

In this thesis we discuss two problems. The first problem is that of Fanno flow in a tube. In [10] the authors have discussed the mathematics of the Fanno model in much more detail than had been previously been done. The analysis in [10] indicates that the Fanno model becomes relevant, if t indicates the unscaled time and t=et , only when t is at least of order O(e- 1) . Indeed, two most important time scales are when t=O(e-1) and t=O(e- 2) . The authors, in the former case, set t=e- 1t1 (t1=t),x=e -11, and obtain the equation 62u6t 21- 62u 6x21=- 2u6 2u6t21 , ( 0.0.1) where u is the velocity of the gas, with p=1,6x1=0 (x1=0). One can follow the solution along the characteristic x1=t1 , and to match with the inviscid behaviour when t1-->0 , u=2+t1 (x1=t1). (0.0.2) In the region t=O(e2) , the authors set t=e2t2, x=e2x2,h= x2t2. For small e , the BC (0.0.02) now becomes u=t2 (x2=t 2), (0.0.3) so that (0.0.1) now has a similarity solution of the form u=t2g( h), u2=e- 1u, and (h2- 1)g'' +4hg'+2g=2g(g+hg' ),' =/ (0.0.4) with g(h)-->2 ash-->1- ,from(0.0.3) (0.0.5) g(h)-->∞ ash-->0- ,(fromthe pressure). ( 0.0.6) In a recent paper [11] the authors discuss the existence of a solution of (0.0.4)-(0.0.6) by using a two dimensional topological shooting method. We also discuss the existence of a solution of (0.0.4)-(0.0.6) by using a shooting method. We first turn the nonlinear ode (0.0.4) into an integral equation and then shoot from the singularity at ∞. The second problem arises when one considers eigenfunction expansions associated with second order ordinary differential equations, as Titchmarsh does in his book. One is concerned with the solutions of the equation - d2ydx2+ q(x)y=ly, (0.0.7) along with certain boundary conditions, where q(x)=-( n2- /)sech 2(x), n=n+/. The problem (0.0.7) has an application in the study of discrete reaction-diffusion equations. Our purpose in this problem is to look in some detail at the equation (0.0.7). We first use contour integrations to give an explicit solution. Secondly, we explore the eigenvalues and the eigenfunctions with the certain boundary conditions. We not only find the spectrum for n = 1, 2, 3, 4 but also explore the eigenvalues for general n where a=0,a=/ and BC is y(x)cosa +y'(x)sin a=0.

A hybrid-perturbation-Galerkin technique which combines multiple expansions

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1989-01-01

A two-step hybrid perturbation-Galerkin method for the solution of a variety of differential equations type problems is found to give better results when multiple perturbation expansions are employed. The method assumes that there is parameter in the problem formulation and that a perturbation method can be sued to construct one or more expansions in this perturbation coefficient functions multiplied by computed amplitudes. In step one, regular and/or singular perturbation methods are used to determine the perturbation coefficient functions. The results of step one are in the form of one or more expansions each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions. In step two the classical Bubnov-Galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes which replace and improve upon the gauge functions. The hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Galerkin methods as applied separately, while combining some of their better features. The proposed method is applied, with two perturbation expansions in each case, to a variety of model ordinary differential equations problems including: a family of linear two-boundary-value problems, a nonlinear two-point boundary-value problem, a quantum mechanical eigenvalue problem and a nonlinear free oscillation problem. The results obtained from the hybrid methods are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.

Ultrarelativistic bound states in the spherical well

DOE Office of Scientific and Technical Information (OSTI.GOV)

Żaba, Mariusz; Garbaczewski, Piotr

2016-07-15

We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−Δ){sup 1/2}, whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral data for lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into non-overlapping, orbitally labelled E{sub (k,l)} series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive lth seriesmore » eigenvalues. Each of them is 2l + 1-degenerate. The l = 0 eigenvalues series E{sub (k,0)} are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E{sub (k,0)}(d = 3) = E{sub 2k}(d = 1). Likewise, the eigenfunctions ψ{sub (k,0)}(d = 3) and ψ{sub 2k}(d = 1) show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Venkatesan, R.C., E-mail: ravi@systemsresearchcorp.com; Plastino, A., E-mail: plastino@fisica.unlp.edu.ar

The (i) reciprocity relations for the relative Fisher information (RFI, hereafter) and (ii) a generalized RFI–Euler theorem are self-consistently derived from the Hellmann–Feynman theorem. These new reciprocity relations generalize the RFI–Euler theorem and constitute the basis for building up a mathematical Legendre transform structure (LTS, hereafter), akin to that of thermodynamics, that underlies the RFI scenario. This demonstrates the possibility of translating the entire mathematical structure of thermodynamics into a RFI-based theoretical framework. Virial theorems play a prominent role in this endeavor, as a Schrödinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI–LTS linkmore » and the quantum mechanical virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf’s, hereafter) and energy-eigenvalues of the above mentioned Schrödinger-like equation. The energy-eigenvalues obtained here via inference are benchmarked against established theoretical and numerical results. A principled theoretical basis to reconstruct the RFI-framework from the FIM framework is established. Numerical examples for exemplary cases are provided. - Highlights: • Legendre transform structure for the RFI is obtained with the Hellmann–Feynman theorem. • Inference of the energy-eigenvalues of the SWE-like equation for the RFI is accomplished. • Basis for reconstruction of the RFI framework from the FIM-case is established. • Substantial qualitative and quantitative distinctions with prior studies are discussed.« less

Eigenvalue Attraction

NASA Astrophysics Data System (ADS)

Movassagh, Ramis

2016-02-01

We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.

Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory

PubMed Central

Eshraghi, Iman; Jalali, Seyed K.; Pugno, Nicola Maria

2016-01-01

Imperfection sensitivity of large amplitude vibration of curved single-walled carbon nanotubes (SWCNTs) is considered in this study. The SWCNT is modeled as a Timoshenko nano-beam and its curved shape is included as an initial geometric imperfection term in the displacement field. Geometric nonlinearities of von Kármán type and nonlocal elasticity theory of Eringen are employed to derive governing equations of motion. Spatial discretization of governing equations and associated boundary conditions is performed using differential quadrature (DQ) method and the corresponding nonlinear eigenvalue problem is iteratively solved. Effects of amplitude and location of the geometric imperfection, and the nonlocal small-scale parameter on the nonlinear frequency for various boundary conditions are investigated. The results show that the geometric imperfection and non-locality play a significant role in the nonlinear vibration characteristics of curved SWCNTs. PMID:28773911

On the acceleration of charged particles at relativistic shock fronts

NASA Technical Reports Server (NTRS)

Kirk, J. G.; Schneider, P.

1987-01-01

The diffusive acceleration of highly relativistic particles at a shock is reconsidered. Using the same physical assumptions as Blandford and Ostriker (1978), but dropping the restriction to nonrelativistic shock velocities, the authors find approximate solutions of the particle kinetic equation by generalizing the diffusion approximation to higher order terms in the anisotropy of the particle distribution. The general solution of the transport equation on either side of the shock is constructed, which involves the solution of an eigenvalue problem. By matching the two solutions at the shock, the spectral index of the resulting power law is found by taking into account a sufficiently large number of eigenfunctions. Low-order truncation corresponds to the standard diffusion approximation and to a somewhat more general method described by Peacock (1981). In addition to the energy spectrum, the method yields the angular distribution of the particles and its spatial dependence.

Conservation laws with coinciding smooth solutions but different conserved variables

NASA Astrophysics Data System (ADS)

Colombo, Rinaldo M.; Guerra, Graziano

2018-04-01

Consider two hyperbolic systems of conservation laws in one space dimension with the same eigenvalues and (right) eigenvectors. We prove that solutions to Cauchy problems with the same initial data differ at third order in the total variation of the initial datum. As a first application, relying on the classical Glimm-Lax result (Glimm and Lax in Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, No. 101. American Mathematical Society, Providence, 1970), we obtain estimates improving those in Saint-Raymond (Arch Ration Mech Anal 155(3):171-199, 2000) on the distance between solutions to the isentropic and non-isentropic inviscid compressible Euler equations, under general equations of state. Further applications are to the general scalar case, where rather precise estimates are obtained, to an approximation by Di Perna of the p-system and to a traffic model.

SIAM Conference on Parallel Processing for Scientific Computing, 4th, Chicago, IL, Dec. 11-13, 1989, Proceedings

NASA Technical Reports Server (NTRS)

Dongarra, Jack (Editor); Messina, Paul (Editor); Sorensen, Danny C. (Editor); Voigt, Robert G. (Editor)

1990-01-01

Attention is given to such topics as an evaluation of block algorithm variants in LAPACK and presents a large-grain parallel sparse system solver, a multiprocessor method for the solution of the generalized Eigenvalue problem on an interval, and a parallel QR algorithm for iterative subspace methods on the CM2. A discussion of numerical methods includes the topics of asynchronous numerical solutions of PDEs on parallel computers, parallel homotopy curve tracking on a hypercube, and solving Navier-Stokes equations on the Cedar Multi-Cluster system. A section on differential equations includes a discussion of a six-color procedure for the parallel solution of elliptic systems using the finite quadtree structure, data parallel algorithms for the finite element method, and domain decomposition methods in aerodynamics. Topics dealing with massively parallel computing include hypercube vs. 2-dimensional meshes and massively parallel computation of conservation laws. Performance and tools are also discussed.

The algebraic theory of latent projectors in lambda matrices

NASA Technical Reports Server (NTRS)

Denman, E. D.; Leyva-Ramos, J.; Jeon, G. J.

1981-01-01

Multivariable systems such as a finite-element model of vibrating structures, control systems, and large-scale systems are often formulated in terms of differential equations which give rise to lambda matrices. The present investigation is concerned with the formulation of the algebraic theory of lambda matrices and the relationship of latent roots, latent vectors, and latent projectors to the eigenvalues, eigenvectors, and eigenprojectors of the companion form. The chain rule for latent projectors and eigenprojectors for the repeated latent root or eigenvalues is given.

Wigner surmises and the two-dimensional homogeneous Poisson point process.

PubMed

Sakhr, Jamal; Nieminen, John M

2006-04-01

We derive a set of identities that relate the higher-order interpoint spacing statistics of the two-dimensional homogeneous Poisson point process to the Wigner surmises for the higher-order spacing distributions of eigenvalues from the three classical random matrix ensembles. We also report a remarkable identity that equates the second-nearest-neighbor spacing statistics of the points of the Poisson process and the nearest-neighbor spacing statistics of complex eigenvalues from Ginibre's ensemble of 2 x 2 complex non-Hermitian random matrices.

Hydrodynamics of the Dirac spectrum

DOE PAGES

Liu, Yizhuang; Warchoł, Piotr; Zahed, Ismail

2015-12-15

We discuss a hydrodynamical description of the eigenvalues of the Dirac spectrum in even dimensions in the vacuum and in the large N (volume) limit. The linearized hydrodynamics supports sound waves. The hydrodynamical relaxation of the eigenvalues is captured by a hydrodynamical (tunneling) minimum configuration which follows from a pertinent form of Euler equation. As a result, the relaxation from a phase of unbroken chiral symmetry to a phase of broken chiral symmetry occurs over a time set by the speed of sound.

Implicit treatment of diffusion terms in lower-upper algorithms

NASA Technical Reports Server (NTRS)

Shih, T. I.-P.; Steinthorsson, E.; Chyu, W. J.

1993-01-01

A method is presented which allows diffusion terms to be treated implicitly in the lower-upper (LU) algorithm (which is a commonly used method for solving 'compressible' Euler and Navier-Stokes equations) so that the algorithm's good stability properties will not be impaired. The new method generalizes the concept of LU factorization from that associated with the sign of eigenvalues to that associated with backward- and forward-difference operators without regard to eigenvalues. The method is verified in a turbulent boundary layer study.

Exact analytic solution of position-dependent mass Schrödinger equation

NASA Astrophysics Data System (ADS)

Rajbongshi, Hangshadhar

2018-03-01

Exact analytic solution of position-dependent mass Schrödinger equation is generated by using extended transformation, a method of mapping a known system into a new system equipped with energy eigenvalues and corresponding wave functions. First order transformation is performed on D-dimensional radial Schrödinger equation with constant mass by taking trigonometric Pöschl-Teller potential as known system. The exactly solvable potentials with position-dependent mass generated for different choices of mass functions through first order transformation are also taken as known systems in the second order transformation performed on D-dimensional radial position-dependent mass Schrödinger equation. The solutions are fitted for "Zhu and Kroemer" ordering of ambiguity. All the wave functions corresponding to nonzero energy eigenvalues are normalizable. The new findings are that the normalizability condition of the wave functions remains independent of mass functions, and some of the generated potentials show a family relationship among themselves where power law potentials also get related to non-power law potentials and vice versa through the transformation.

Reliable use of determinants to solve nonlinear structural eigenvalue problems efficiently

NASA Technical Reports Server (NTRS)

Williams, F. W.; Kennedy, D.

1988-01-01

The analytical derivation, numerical implementation, and performance of a multiple-determinant parabolic interpolation method (MDPIM) for use in solving transcendental eigenvalue (critical buckling or undamped free vibration) problems in structural mechanics are presented. The overall bounding, eigenvalue-separation, qualified parabolic interpolation, accuracy-confirmation, and convergence-recovery stages of the MDPIM are described in detail, and the numbers of iterations required to solve sample plane-frame problems using the MDPIM are compared with those for a conventional bisection method and for the Newtonian method of Simpson (1984) in extensive tables. The MDPIM is shown to use 31 percent less computation time than bisection when accuracy of 0.0001 is required, but 62 percent less when accuracy of 10 to the -8th is required; the time savings over the Newtonian method are about 10 percent.

A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems

DOE PAGES

Li, Ruipeng; Xi, Yuanzhe; Vecharynski, Eugene; ...

2016-08-16

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a thick-restart version of the Lanczos algorithm with deflation ("locking'') and a new type of polynomial filter obtained from a least-squares technique. Furthermore, the resulting algorithm can be utilized in a “spectrum-slicing” approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different subintervals independently from onemore » another.« less

Persistence in a Two-Dimensional Moving-Habitat Model.

PubMed

Phillips, Austin; Kot, Mark

2015-11-01

Environmental changes are forcing many species to track suitable conditions or face extinction. In this study, we use a two-dimensional integrodifference equation to analyze whether a population can track a habitat that is moving due to climate change. We model habitat as a simple rectangle. Our model quickly leads to an eigenvalue problem that determines whether the population persists or declines. After surveying techniques to solve the eigenvalue problem, we highlight three findings that impact conservation efforts such as reserve design and species risk assessment. First, while other models focus on habitat length (parallel to the direction of habitat movement), we show that ignoring habitat width (perpendicular to habitat movement) can lead to overestimates of persistence. Dispersal barriers and hostile landscapes that constrain habitat width greatly decrease the population's ability to track its habitat. Second, for some long-distance dispersal kernels, increasing habitat length improves persistence without limit; for other kernels, increasing length is of limited help and has diminishing returns. Third, it is not always best to orient the long side of the habitat in the direction of climate change. Evidence suggests that the kurtosis of the dispersal kernel determines whether it is best to have a long, wide, or square habitat. In particular, populations with platykurtic dispersal benefit more from a wide habitat, while those with leptokurtic dispersal benefit more from a long habitat. We apply our model to the Rocky Mountain Apollo butterfly (Parnassius smintheus).

Tidal frequency estimation for closed basins

NASA Technical Reports Server (NTRS)

Eades, J. B., Jr.

1978-01-01

A method was developed for determining the fundamental tidal frequencies for closed basins of water, by means of an eigenvalue analysis. The mathematical model employed, was the Laplace tidal equations.

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

NASA Astrophysics Data System (ADS)

Liu, Jianzhou; Wang, Li; Zhang, Juan

2017-11-01

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.

Fluid displacement between two parallel plates: a non-empirical model displaying change of type from hyperbolic to elliptic equations

NASA Astrophysics Data System (ADS)

Shariati, M.; Talon, L.; Martin, J.; Rakotomalala, N.; Salin, D.; Yortsos, Y. C.

2004-11-01

We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as ‘three-fluid’ flow in the same geometry. Assuming symmetry across the gap and based on the lubrication (‘equilibrium’) approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations. The hyperbolic-to-elliptic change-of-type reflects the failing of the lubrication approximation, underlying the quasi-linear hyperbolic formalism, to describe the problem uniformly. The obtained solution is analogous to non-classical shocks recently suggested in problems with change of type.

A stability analysis of the power-law steady state of marine size spectra.

PubMed

Datta, Samik; Delius, Gustav W; Law, Richard; Plank, Michael J

2011-10-01

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a "jump-growth" equation, a first order approximation which is the widely used McKendrick-von Foerster equation, and a second order approximation which is the McKendrick-von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick-von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick-von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator:prey mass ratio, a large diet breadth and a high feeding efficiency. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.

A case against a divide and conquer approach to the nonsymmetric eigenvalue problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1991-12-01

Divide and conquer techniques based on rank-one updating have proven fast, accurate, and efficient in parallel for the real symmetric tridiagonal and unitary eigenvalue problems and for the bidiagonal singular value problem. Although the divide and conquer mechanism can also be adapted to the real nonsymmetric eigenproblem in a straightforward way, most of the desirable characteristics of the other algorithms are lost. In this paper, we examine the problems of accuracy and efficiency that can stand in the way of a nonsymmetric divide and conquer eigensolver based on low-rank updating. 31 refs., 2 figs.

Mean Flow Augmented Acoustics in Rocket Systems

NASA Technical Reports Server (NTRS)

Fischbach, Sean R.

2014-01-01

Oscillatory motion in solid rocket motors and liquid engines has long been a subject of concern. Many rockets display violent fluctuations in pressure, velocity, and temperature originating from the complex interactions between the combustion process and gas dynamics. The customary approach to modeling acoustic waves inside a rocket chamber is to apply the classical inhomogeneous wave equation to the combustion gas. The assumption of a linear, non-dissipative wave in a quiescent fluid remains valid while the acoustic amplitudes are small and local gas velocities stay below Mach 0.2. The converging section of a rocket nozzle, where gradients in pressure, density, and velocity become large, is a notable region where this approach is not applicable. The expulsion of unsteady energy through the nozzle of a rocket is identified as the predominate source of acoustic damping for most rocket systems. An accurate model of the acoustic behavior within this region where acoustic modes are influenced by the presence of a steady mean flow is required for reliable stability predictions. Recently, an approach to address nozzle damping with mean flow effects was implemented by French [1]. This new approach extends the work originated by Sigman and Zinn [2] by solving the acoustic velocity potential equation (AVPE) formulated by perturbing the Euler equations [3]. The acoustic velocity potential (psi) describing the acoustic wave motion in the presence of an inhomogeneous steady high-speed flow is defined by, (del squared)(psi) - (lambda/c)(exp 2)(psi) - M(dot)[M(dot)(del)(del(psi))] - 2(lambda(M/c) + (M(dot)del(M))(dot)del(psi)-2(lambda)(psi)[M(dot)del(1/c)]=0 (1) with M as the Mach vector, c as the speed of sound, and lambda as the complex eigenvalue. French apply the finite volume method to solve the steady flow field within the combustion chamber and nozzle with inviscid walls. The complex eigenvalues and eigenvector are determined with the use of the ARPACK eigensolver. The present study employs the COMSOL Multphysics framework to solve the coupled eigenvalue problem using the finite element approach. The study requires one way coupling of the CFD High Mach Number Flow (HMNF) and mathematics module. The HMNF module evaluated the gas flow inside of a solid rocket motor using St. Robert's law modeling solid propellant burn rate, slip boundary conditions, and the supersonic outflow condition. Results from the HMNF model are used by the coefficient form of the mathematics module to determine the eigenvalues of the AVPE. The mathematics model is truncated at the nozzle sonic line, where a zero flux boundary condition is self-satisfying. The remaining boundaries are modeled with a zero flux boundary condition, assuming zero acoustic absorption on all surfaces. Pertinent results from these analyses are the complex valued eigenvalue and eigenvectors. Comparisons are made to the French results to evaluate the modeling approach. A comparison of the French results with that of the present analysis is displayed in figures 1 and 2, respectively. The graphic shows the first tangential eigenvector's real (a) and imaginary (b) values.

Gauss Seidel-type methods for energy states of a multi-component Bose Einstein condensate

NASA Astrophysics Data System (ADS)

Chang, Shu-Ming; Lin, Wen-Wei; Shieh, Shih-Feng

2005-01-01

In this paper, we propose two iterative methods, a Jacobi-type iteration (JI) and a Gauss-Seidel-type iteration (GSI), for the computation of energy states of the time-independent vector Gross-Pitaevskii equation (VGPE) which describes a multi-component Bose-Einstein condensate (BEC). A discretization of the VGPE leads to a nonlinear algebraic eigenvalue problem (NAEP). We prove that the GSI method converges locally and linearly to a solution of the NAEP if and only if the associated minimized energy functional problem has a strictly local minimum. The GSI method can thus be used to compute ground states and positive bound states, as well as the corresponding energies of a multi-component BEC. Numerical experience shows that the GSI converges much faster than JI and converges globally within 10-20 steps.

Discontinuous Galerkin Methods for NonLinear Differential Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy; Mansour, Nagi (Technical Monitor)

2001-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hintermueller, M., E-mail: hint@math.hu-berlin.de; Kao, C.-Y., E-mail: Ckao@claremontmckenna.edu; Laurain, A., E-mail: laurain@math.hu-berlin.de

2012-02-15

This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this thresholdmore » value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed Robin-Neumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement.« less

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

NASA Astrophysics Data System (ADS)

Lakshtanov, E.; Vainberg, B.

2013-10-01

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on a possible location of the transmission eigenvalues. If the index of refraction \\sqrt{n(x)} is real, then we obtain a result on the existence of infinitely many positive ITEs and the Weyl-type lower bound on its counting function. All the results are obtained under the assumption that n(x) - 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).

Crossflow effects on the growth rate of inviscid Goertler vortices in a hypersonic boundary layer

NASA Technical Reports Server (NTRS)

Fu, Yibin; Hall, Philip

1992-01-01

The effects of crossflow on the growth rate of inviscid Goertler vortices in a hypersonic boundary layer with pressure gradient are studied. Attention is focused on the inviscid mode trapped in the temperature adjustment layer; this mode has greater growth rate than any other mode. The eigenvalue problem which governs the relationship between the growth rate, the crossflow amplitude, and the wavenumber is solved numerically, and the results are then used to clarify the effects of crossflow on the growth rate of inviscid Goertler vortices. It is shown that crossflow effects on Goertler vortices are fundamentally different for incompressible and hypersonic flows. The neutral mode eigenvalue problem is found to have an exact solution, and as a by-product, we have also found the exact solution to a neutral mode eigenvalue problem which was formulated, but unsolved before, by Bassom and Hall (1991).

Deterministically estimated fission source distributions for Monte Carlo k-eigenvalue problems

DOE PAGES

Biondo, Elliott D.; Davidson, Gregory G.; Pandya, Tara M.; ...

2018-04-30

The standard Monte Carlo (MC) k-eigenvalue algorithm involves iteratively converging the fission source distribution using a series of potentially time-consuming inactive cycles before quantities of interest can be tallied. One strategy for reducing the computational time requirements of these inactive cycles is the Sourcerer method, in which a deterministic eigenvalue calculation is performed to obtain an improved initial guess for the fission source distribution. This method has been implemented in the Exnihilo software suite within SCALE using the SPNSPN or SNSN solvers in Denovo and the Shift MC code. The efficacy of this method is assessed with different Denovo solutionmore » parameters for a series of typical k-eigenvalue problems including small criticality benchmarks, full-core reactors, and a fuel cask. Here it is found that, in most cases, when a large number of histories per cycle are required to obtain a detailed flux distribution, the Sourcerer method can be used to reduce the computational time requirements of the inactive cycles.« less

Deterministically estimated fission source distributions for Monte Carlo k-eigenvalue problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Biondo, Elliott D.; Davidson, Gregory G.; Pandya, Tara M.

The standard Monte Carlo (MC) k-eigenvalue algorithm involves iteratively converging the fission source distribution using a series of potentially time-consuming inactive cycles before quantities of interest can be tallied. One strategy for reducing the computational time requirements of these inactive cycles is the Sourcerer method, in which a deterministic eigenvalue calculation is performed to obtain an improved initial guess for the fission source distribution. This method has been implemented in the Exnihilo software suite within SCALE using the SPNSPN or SNSN solvers in Denovo and the Shift MC code. The efficacy of this method is assessed with different Denovo solutionmore » parameters for a series of typical k-eigenvalue problems including small criticality benchmarks, full-core reactors, and a fuel cask. Here it is found that, in most cases, when a large number of histories per cycle are required to obtain a detailed flux distribution, the Sourcerer method can be used to reduce the computational time requirements of the inactive cycles.« less

Quantization with maximally degenerate Poisson brackets: the harmonic oscillator!

NASA Astrophysics Data System (ADS)

Nutku, Yavuz

2003-07-01

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions, which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single-valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.

A Relaxation Method for Nonlocal and Non-Hermitian Operators

NASA Astrophysics Data System (ADS)

Lagaris, I. E.; Papageorgiou, D. G.; Braun, M.; Sofianos, S. A.

1996-06-01

We present a grid method to solve the time dependent Schrödinger equation (TDSE). It uses the Crank-Nicholson scheme to propagate the wavefunction forward in time and finite differences to approximate the derivative operators. The resulting sparse linear system is solved by the symmetric successive overrelaxation iterative technique. The method handles local and nonlocal interactions and Hamiltonians that correspond to either Hermitian or to non-Hermitian matrices with real eigenvalues. We test the method by solving the TDSE in the imaginary time domain, thus converting the time propagation to asymptotic relaxation. Benchmark problems solved are both in one and two dimensions, with local, nonlocal, Hermitian and non-Hermitian Hamiltonians.

Linear static structural and vibration analysis on high-performance computers

NASA Technical Reports Server (NTRS)

Baddourah, M. A.; Storaasli, O. O.; Bostic, S. W.

1993-01-01

Parallel computers offer the oppurtunity to significantly reduce the computation time necessary to analyze large-scale aerospace structures. This paper presents algorithms developed for and implemented on massively-parallel computers hereafter referred to as Scalable High-Performance Computers (SHPC), for the most computationally intensive tasks involved in structural analysis, namely, generation and assembly of system matrices, solution of systems of equations and calculation of the eigenvalues and eigenvectors. Results on SHPC are presented for large-scale structural problems (i.e. models for High-Speed Civil Transport). The goal of this research is to develop a new, efficient technique which extends structural analysis to SHPC and makes large-scale structural analyses tractable.

Making sense of quantum operators, eigenstates and quantum measurements

NASA Astrophysics Data System (ADS)

Gire, Elizabeth; Manogue, Corinne

2012-02-01

Operators play a central role in the formalism of quantum mechanics. In particular, operators corresponding to observables encode important information about the results of quantum measurements. We interviewed upper-level undergraduate physics majors about their understanding of the role of operators in quantum measurements. Previous studies have shown that many students think of measurements on quantum systems as being deterministic and that measurements mathematically correspond to operators acting on the initial quantum state. This study is consistent with and expands on those results. We report on how two students make sense of a quantum measurement problem involving sequential measurements and the role that the eigenvalue equation plays in this sense-making.

Inertial modes in a rotating triaxial ellipsoid

PubMed Central

Vantieghem, S.

2014-01-01

In this work, we present an algorithm that enables computation of inertial modes and their corresponding frequencies in a rotating triaxial ellipsoid. The method consists of projecting the inertial mode equation onto finite-dimensional bases of polynomial vector fields. It is shown that this leads to a well-posed eigenvalue problem, and hence, that eigenmodes are of polynomial form. Furthermore, these results shed new light onto the question whether the eigenmodes form a complete basis, i.e. whether any arbitrary velocity field can be expanded in a sum of inertial modes. Finally, we prove that two intriguing integral properties of inertial modes in rotating spheres and spheroids also extend to triaxial ellipsoids. PMID:25104908

Proper Orthogonal Decomposition in Optimal Control of Fluids

NASA Technical Reports Server (NTRS)

Ravindran, S. S.

1999-01-01

In this article, we present a reduced order modeling approach suitable for active control of fluid dynamical systems based on proper orthogonal decomposition (POD). The rationale behind the reduced order modeling is that numerical simulation of Navier-Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. We examine the possibility of obtaining reduced order models that reduce computational complexity associated with the Navier-Stokes equations while capturing the essential dynamics by using the POD. The POD allows extraction of certain optimal set of basis functions, perhaps few, from a computational or experimental data-base through an eigenvalue analysis. The solution is then obtained as a linear combination of these optimal set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations. We here use it in active control of fluid flows governed by the Navier-Stokes equations. We show that the resulting reduced order model can be very efficient for the computations of optimization and control problems in unsteady flows. Finally, implementational issues and numerical experiments are presented for simulations and optimal control of fluid flow through channels.

A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations

NASA Astrophysics Data System (ADS)

Chen, Hao; Lv, Wen; Zhang, Tongtong

2018-05-01

We study preconditioned iterative methods for the linear system arising in the numerical discretization of a two-dimensional space-fractional diffusion equation. Our approach is based on a formulation of the discrete problem that is shown to be the sum of two Kronecker products. By making use of an alternating Kronecker product splitting iteration technique we establish a class of fixed-point iteration methods. Theoretical analysis shows that the new method converges to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real. The basic iteration is accelerated by a Krylov subspace method like GMRES. The corresponding preconditioner is in a form of a Kronecker product structure and requires at each iteration the solution of a set of discrete one-dimensional fractional diffusion equations. We use structure preserving approximations to the discrete one-dimensional fractional diffusion operators in the action of the preconditioning matrix. Numerical examples are presented to illustrate the effectiveness of this approach.

Gaussian step-pressure loading of rigid viscoplastic plates. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Hayduk, R. J.; Durling, B. J.

1978-01-01

The response of a thin, rigid viscoplastic plate subjected to a spatially axisymmetric Gaussian step pressure impulse loading was studied analytically. A Gaussian pressure distribution in excess of the collapse load was applied to the plate, held constant for a length of time, and then suddenly removed. The plate deforms with monotonically increasing deflections until the dynamic energy is completely dissipated in plastic work. The simply supported plate of uniform thickness obeys the von Mises yield criterion and a generalized constitutive equation for rigid viscoplastic materials. For the small deflection bending response of the plate, the governing system of equations is essentially nonlinear. Transverse shear stress is neglected in the yield condition and rotary inertia in the equations of dynamic equilibrium. A proportional loading technique, known to give excellent approximations of the exact solution for the uniform load case, was used to linearize the problem and to obtain the analytical solutions in the form of eigenvalue expansions. The effects of load concentration, of an order of magnitude change in the viscosity of the plate material, and of load duration were examined while holding the total impulse constant.

On the r-mode spectrum of relativistic stars in the low-frequency approximation

NASA Astrophysics Data System (ADS)

Ruoff, Johannes; Kokkotas, Kostas D.

2001-12-01

The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time-independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for l=2, it is nevertheless also possible to find discrete mode solutions (the r modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time-dependent equations. For stellar models admitting a physical r-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of l it seems that in certain cases there are no mode solutions at all.

CAVE: A computer code for two-dimensional transient heating analysis of conceptual thermal protection systems for hypersonic vehicles

NASA Technical Reports Server (NTRS)

Rathjen, K. A.

1977-01-01

A digital computer code CAVE (Conduction Analysis Via Eigenvalues), which finds application in the analysis of two dimensional transient heating of hypersonic vehicles is described. The CAVE is written in FORTRAN 4 and is operational on both IBM 360-67 and CDC 6600 computers. The method of solution is a hybrid analytical numerical technique that is inherently stable permitting large time steps even with the best of conductors having the finest of mesh size. The aerodynamic heating boundary conditions are calculated by the code based on the input flight trajectory or can optionally be calculated external to the code and then entered as input data. The code computes the network conduction and convection links, as well as capacitance values, given basic geometrical and mesh sizes, for four generations (leading edges, cooled panels, X-24C structure and slabs). Input and output formats are presented and explained. Sample problems are included. A brief summary of the hybrid analytical-numerical technique, which utilizes eigenvalues (thermal frequencies) and eigenvectors (thermal mode vectors) is given along with aerodynamic heating equations that have been incorporated in the code and flow charts.

Analyzing Aeroelastic Stability of a Tilt-Rotor Aircraft

NASA Technical Reports Server (NTRS)

Kvaternil, Raymond G.

2006-01-01

Proprotor Aeroelastic Stability Analysis, now at version 4.5 (PASTA 4.5), is a FORTRAN computer program for analyzing the aeroelastic stability of a tiltrotor aircraft in the airplane mode of flight. The program employs a 10-degree- of-freedom (DOF), discrete-coordinate, linear mathematical model of a rotor with three or more blades and its drive system coupled to a 10-DOF modal model of an airframe. The user can select which DOFs are included in the analysis. Quasi-steady strip-theory aerodynamics is employed for the aerodynamic loads on the blades, a quasi-steady representation is employed for the aerodynamic loads acting on the vibrational modes of the airframe, and a stability-derivative approach is used for the aerodynamics associated with the rigid-body DOFs of the airframe. Blade parameters that vary with the blade collective pitch can be obtained by interpolation from a user-defined table. Stability is determined by examining the eigenvalues that are obtained by solving the coupled equations of motions as a matrix eigenvalue problem. Notwithstanding the relative simplicity of its mathematical foundation, PASTA 4.5 and its predecessors have played key roles in a number of engineering investigations over the years.

Parametric study of the dynamic JWL-EOS for detonation products

DOE Office of Scientific and Technical Information (OSTI.GOV)

Urtiew, P.A.; Hayes, B.

1990-03-01

The JWL equation of state describing the adiabatic expansion of detonation products is revisited to complete the description of the principal eigenvalue, to reset the secondary eigenvalue to produce a well-behaved adiabatic gamma profile, and to normalize the characteristic equation of state in terms of conventional parameters having a clear experimental interpretation. This is accomplished by interjecting a dynamic flow condition concerning the value of the relative specific volume when the particle velocity of the detonation products is zero. In addition, a set of generic parameters based on the statistical distribution of the primary explosives making up the available datamore » base is presented. Unlike theoretical and statistical mechanical models, the adiabatic gamma function for these materials is seen to have a positive initial slope in accord with experimental findings. 10 refs., 4 figs.« less

Methods for the calculation of axial wave numbers in lined ducts with mean flow

NASA Technical Reports Server (NTRS)

Eversman, W.

1981-01-01

A survey is made of the methods available for the calculation of axial wave numbers in lined ducts. Rectangular and circular ducts with both uniform and non-uniform flow are considered as are ducts with peripherally varying liners. A historical perspective is provided by a discussion of the classical methods for computing attenuation when no mean flow is present. When flow is present these techniques become either impractical or impossible. A number of direct eigenvalue determination schemes which have been used when flow is present are discussed. Methods described are extensions of the classical no-flow technique, perturbation methods based on the no-flow technique, direct integration methods for solution of the eigenvalue equation, an integration-iteration method based on the governing differential equation for acoustic transmission, Galerkin methods, finite difference methods, and finite element methods.

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: An eigenvalue analysis

NASA Technical Reports Server (NTRS)

Warming, Robert F.; Beam, Richard M.

1986-01-01

A hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.

Sparse Regression as a Sparse Eigenvalue Problem

NASA Technical Reports Server (NTRS)

Moghaddam, Baback; Gruber, Amit; Weiss, Yair; Avidan, Shai

2008-01-01

We extend the l0-norm "subspectral" algorithms for sparse-LDA [5] and sparse-PCA [6] to general quadratic costs such as MSE in linear (kernel) regression. The resulting "Sparse Least Squares" (SLS) problem is also NP-hard, by way of its equivalence to a rank-1 sparse eigenvalue problem (e.g., binary sparse-LDA [7]). Specifically, for a general quadratic cost we use a highly-efficient technique for direct eigenvalue computation using partitioned matrix inverses which leads to dramatic x103 speed-ups over standard eigenvalue decomposition. This increased efficiency mitigates the O(n4) scaling behaviour that up to now has limited the previous algorithms' utility for high-dimensional learning problems. Moreover, the new computation prioritizes the role of the less-myopic backward elimination stage which becomes more efficient than forward selection. Similarly, branch-and-bound search for Exact Sparse Least Squares (ESLS) also benefits from partitioned matrix inverse techniques. Our Greedy Sparse Least Squares (GSLS) generalizes Natarajan's algorithm [9] also known as Order-Recursive Matching Pursuit (ORMP). Specifically, the forward half of GSLS is exactly equivalent to ORMP but more efficient. By including the backward pass, which only doubles the computation, we can achieve lower MSE than ORMP. Experimental comparisons to the state-of-the-art LARS algorithm [3] show forward-GSLS is faster, more accurate and more flexible in terms of choice of regularization

The method of fundamental solutions for computing acoustic interior transmission eigenvalues

NASA Astrophysics Data System (ADS)

Kleefeld, Andreas; Pieronek, Lukas

2018-03-01

We analyze the method of fundamental solutions (MFS) in two different versions with focus on the computation of approximate acoustic interior transmission eigenvalues in 2D for homogeneous media. Our approach is mesh- and integration free, but suffers in general from the ill-conditioning effects of the discretized eigenoperator, which we could then successfully balance using an approved stabilization scheme. Our numerical examples cover many of the common scattering objects and prove to be very competitive in accuracy with the standard methods for PDE-related eigenvalue problems. We finally give an approximation analysis for our framework and provide error estimates, which bound interior transmission eigenvalue deviations in terms of some generalized MFS output.

Chaotic dynamics and diffusion in a piecewise linear equation

NASA Astrophysics Data System (ADS)

Shahrear, Pabel; Glass, Leon; Edwards, Rod

2015-03-01

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

Time-periodic solutions of the Benjamin-Ono equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ambrose , D.M.; Wilkening, Jon

2008-04-01

We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one ofmore » the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.« less

Stability analysis of rimming flow inside a horizontally rotating cylinder in the presence of an insoluble surfactant

NASA Astrophysics Data System (ADS)

Kumawat, Tara Chand; Tiwari, Naveen

2017-12-01

Two-dimensional base state solutions for rimming flows and their stability analysis to small axial perturbations are analyzed numerically. A thin liquid film which is uniformly covered with an insoluble surfactant flows inside a counterclockwise rotating horizontal cylinder. In the present work, a mathematical model is obtained which consists of coupled thin film thickness and surfactant concentration evolution equations. The governing equations are obtained by simplifying the momentum and species transport equations using the thin-film approximation. The model equations include the effect of gravity, viscosity, capillarity, inertia, and Marangoni stress. The concentration gradients generated due to flow result in the surface tension gradient that generates the Marangoni stress near the interface region. The oscillations in the flow due to inertia are damped out by the Marangoni stress. It is observed that the Marangoni stress has stabilizing effect, whereas inertia and surface tension enhance the instability growth rate. In the presence of low diffusion of the surfactant or large value of the Péclet number, the Marangoni stress becomes more effective. The analytically obtained eigenvalues match well with the numerically computed eigenvalues in the absence of gravity.

Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry

NASA Astrophysics Data System (ADS)

Besse, Nicolas; Coulette, David

2016-08-01

Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov-Poisson and Vlasov-Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, "Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry" (submitted)] and were found to be surprisingly close to those for the original gyrokinetic-Vlasov equations. The purpose of the present paper is to make these new ideas accessible to two readerships: applied mathematicians and plasma physicists.

Approximate controllability of a system of parabolic equations with delay

NASA Astrophysics Data System (ADS)

Carrasco, Alexander; Leiva, Hugo

2008-09-01

In this paper we give necessary and sufficient conditions for the approximate controllability of the following system of parabolic equations with delay: where [Omega] is a bounded domain in , D is an n×n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the control and B[set membership, variant]L(U,Z) with , . The standard notation zt(x) defines a function from [-[tau],0] to (with x fixed) by zt(x)(s)=z(t+s,x), -[tau][less-than-or-equals, slant]s[less-than-or-equals, slant]0. Here [tau][greater-or-equal, slanted]0 is the maximum delay, which is supposed to be finite. We assume that the operator is linear and bounded, and [phi]0[set membership, variant]Z, [phi][set membership, variant]L2([-[tau],0];Z). To this end: First, we reformulate this system into a standard first-order delay equation. Secondly, the semigroup associated with the first-order delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the Laplacian operator (); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the approximate controllability of delay system to derive our main result.

Kinetic energy partition method applied to ground state helium-like atoms.

PubMed

Chen, Yu-Hsin; Chao, Sheng D

2017-03-28

We have used the recently developed kinetic energy partition (KEP) method to solve the quantum eigenvalue problems for helium-like atoms and obtain precise ground state energies and wave-functions. The key to treating properly the electron-electron (repulsive) Coulomb potential energies for the KEP method to be applied is to introduce a "negative mass" term into the partitioned kinetic energy. A Hartree-like product wave-function from the subsystem wave-functions is used to form the initial trial function, and the variational search for the optimized adiabatic parameters leads to a precise ground state energy. This new approach sheds new light on the all-important problem of solving many-electron Schrödinger equations and hopefully opens a new way to predictive quantum chemistry. The results presented here give very promising evidence that an effective one-electron model can be used to represent a many-electron system, in the spirit of density functional theory.

Mechanics of composite materials: Recent advances; Proceedings of the Symposium, Virginia Polytechnic Institute and State University, Blacksburg, VA, August 16-19, 1982

NASA Technical Reports Server (NTRS)

Hashin, Z. (Editor); Herakovich, C. T. (Editor)

1983-01-01

The present conference on the mechanics of composites discusses microstructure's influence on particulate and short fiber composites' thermoelastic and transport properties, the elastoplastic deformation of composites, constitutive equations for viscoplastic composites, the plasticity and fatigue of metal matrix composites, laminate damping mechanisms, the micromechanical modeling of Kevlar/epoxy composites' time-dependent failure, the variational characterization of waves in composites, and computational methods for eigenvalue problems in composite design. Also discussed are the elastic response of laminates, elastic coupling nonlinear effects in unsymmetrical laminates, elasticity solutions for laminate problems having stress singularities, the mechanics of bimodular composite structures, the optimization of laminated plates and shells, NDE for laminates, the role of matrix cracking in the continuum constitutive behavior of a damaged composite ply, and the energy release rates of various microcracks in short fiber composites.

Designing pinhole vacancies in graphene towards functionalization: Effects on critical buckling load

NASA Astrophysics Data System (ADS)

Georgantzinos, S. K.; Markolefas, S.; Giannopoulos, G. I.; Katsareas, D. E.; Anifantis, N. K.

2017-03-01

The effect of size and placement of pinhole-type atom vacancies on Euler's critical load on free-standing, monolayer graphene, is investigated. The graphene is modeled by a structural spring-based finite element approach, in which every interatomic interaction is approached as a linear spring. The geometry of graphene and the pinhole size lead to the assembly of the stiffness matrix of the nanostructure. Definition of the boundary conditions of the problem leads to the solution of the eigenvalue problem and consequently to the critical buckling load. Comparison to results found in the literature illustrates the validity and accuracy of the proposed method. Parametric analysis regarding the placement and size of the pinhole-type vacancy, as well as the graphene geometry, depicts the effects on critical buckling load. Non-linear regression analysis leads to empirical-analytical equations for predicting the buckling behavior of graphene, with engineered pinhole-type atom vacancies.

Accuracy of analytic energy level formulas applied to hadronic spectroscopy of heavy mesons

NASA Technical Reports Server (NTRS)

Badavi, Forooz F.; Norbury, John W.; Wilson, John W.; Townsend, Lawrence W.

1988-01-01

Linear and harmonic potential models are used in the nonrelativistic Schroedinger equation to obtain article mass spectra for mesons as bound states of quarks. The main emphasis is on the linear potential where exact solutions of the S-state eigenvalues and eigenfunctions and the asymptotic solution for the higher order partial wave are obtained. A study of the accuracy of two analytical energy level formulas as applied to heavy mesons is also included. Cornwall's formula is found to be particularly accurate and useful as a predictor of heavy quarkonium states. Exact solution for all partial waves of eigenvalues and eigenfunctions for a harmonic potential is also obtained and compared with the calculated discrete spectra of the linear potential. Detailed derivations of the eigenvalues and eigenfunctions of the linear and harmonic potentials are presented in appendixes.

Analysis of Eigenvalue and Eigenfunction of Klein Gordon Equation Using Asymptotic Iteration Method for Separable Non-central Cylindrical Potential

NASA Astrophysics Data System (ADS)

Suparmi, A.; Cari, C.; Lilis Elviyanti, Isnaini

2018-04-01

Analysis of relativistic energy and wave function for zero spin particles using Klein Gordon equation was influenced by separable noncentral cylindrical potential was solved by asymptotic iteration method (AIM). By using cylindrical coordinates, the Klein Gordon equation for the case of symmetry spin was reduced to three one-dimensional Schrodinger like equations that were solvable using variable separation method. The relativistic energy was calculated numerically with Matlab software, and the general unnormalized wave function was expressed in hypergeometric terms.

Simple derivation of the Lindblad equation

NASA Astrophysics Data System (ADS)

Pearle, Philip

2012-07-01

The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is ‘simple’ in that all it uses is the expression of a Hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first given in a two-dimensional Hilbert space.

Frictionless Contact of Multilayered Composite Half Planes Containing Layers With Complex Eigenvalues

NASA Technical Reports Server (NTRS)

Zhang, Wang; Binienda, Wieslaw K.; Pindera, Marek-Jerzy

1997-01-01

A previously developed local-global stiffness matrix methodology for the response of a composite half plane, arbitrarily layered with isotropic, orthotropic or monoclinic plies, to indentation by a rigid parabolic punch is further extended to accommodate the presence of layers with complex eigenvalues (e.g., honeycomb or piezoelectric layers). First, a generalized plane deformation solution for the displacement field in an orthotropic layer or half plane characterized by complex eigenvalues is obtained using Fourier transforms. A local stiffness matrix in the transform domain is subsequently constructed for this class of layers and half planes, which is then assembled into a global stiffness matrix for the entire multilayered half plane by enforcing continuity conditions along the interfaces. Application of the mixed boundary condition on the top surface of the half plane indented by a rigid punch results in an integral equation for the unknown pressure in the contact region. The integral possesses a divergent kernel which is decomposed into Cauchy-type and regular parts using the asymptotic properties of the local stiffness matrix and a relationship between Fourier and finite Hilbert transform of the contact pressure. The solution of the resulting singular integral equation is obtained using a collocation technique based on the properties of orthogonal polynomials developed by Erdogan and Gupta. Examples are presented that illustrate the important influence of low transverse properties of layers with complex eigenvalues, such as those exhibited by honeycomb, on the load versus contact length response and contact pressure distributions for half planes containing typical composite materials.

Dynamic Eigenvalue Problem of Concrete Slab Road Surface

NASA Astrophysics Data System (ADS)

Pawlak, Urszula; Szczecina, Michał

2017-10-01

The paper presents an analysis of the dynamic eigenvalue problem of concrete slab road surface. A sample concrete slab was modelled using Autodesk Robot Structural Analysis software and calculated with Finite Element Method. The slab was set on a one-parameter elastic subsoil, for which the modulus of elasticity was separately calculated. The eigen frequencies and eigenvectors (as maximal vertical nodal displacements) were presented. On the basis of the results of calculations, some basic recommendations for designers of concrete road surfaces were offered.

Density-matrix-based algorithm for solving eigenvalue problems

NASA Astrophysics Data System (ADS)

Polizzi, Eric

2009-03-01

A fast and stable numerical algorithm for solving the symmetric eigenvalue problem is presented. The technique deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques and takes its inspiration from the contour integration and density-matrix representation in quantum mechanics. It will be shown that this algorithm—named FEAST—exhibits high efficiency, robustness, accuracy, and scalability on parallel architectures. Examples from electronic structure calculations of carbon nanotubes are presented, and numerical performances and capabilities are discussed.

Eigenvalue Problems.

DTIC Science & Technology

1987-06-01

Vibration of an Elastic Bar We are interested in studying the small, longitudinal vibra- tions of a longitudinally loaded, elastically supported, elastic...u 2 + + 2u O(( m,Q Uk .(J- MO In the study of eigenvalue problems, central use will be made of Rellich’s theorem (cf. Agmon [19651), which states...H , where a > 0. Sufficient conditions for (4.2) - (4.4) to hold were given in Section 3; cf. (3.15) -(3.17). For the study of (4.1) it is useful to

Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

HyeongKae Park; Robert Nourgaliev; Vincent Mousseau

2008-07-01

A novel numerical algorithm (rDG-JFNK) for all-speed fluid flows with heat conduction and viscosity is introduced. The rDG-JFNK combines the Discontinuous Galerkin spatial discretization with the implicit Runge-Kutta time integration under the Jacobian-free Newton-Krylov framework. We solve fully-compressible Navier-Stokes equations without operator-splitting of hyperbolic, diffusion and reaction terms, which enables fully-coupled high-order temporal discretization. The stability constraint is removed due to the L-stable Explicit, Singly Diagonal Implicit Runge-Kutta (ESDIRK) scheme. The governing equations are solved in the conservative form, which allows one to accurately compute shock dynamics, as well as low-speed flows. For spatial discretization, we develop a “recovery” familymore » of DG, exhibiting nearly-spectral accuracy. To precondition the Krylov-based linear solver (GMRES), we developed an “Operator-Split”-(OS) Physics Based Preconditioner (PBP), in which we transform/simplify the fully-coupled system to a sequence of segregated scalar problems, each can be solved efficiently with Multigrid method. Each scalar problem is designed to target/cluster eigenvalues of the Jacobian matrix associated with a specific physics.« less

The Coupling between Earth's Inertial and Rotational Eigenmodes

NASA Astrophysics Data System (ADS)

Triana, S. A.; Rekier, J.; Trinh, A.; Laguerre, R.; Zhu, P.; Dehant, V. M. A.

2017-12-01

Wave motions in the Earth's fluid core, supported by the restoring action of both buoyancy (within the stably stratified top layer) and the Coriolis force, lead to the existence of global oscillation modes, the so-called gravito-inertial modes. These fluid modes can couple with the rotational modes of the Earth by exerting torques on the mantle and the inner core. Viscous shear stresses at the fluid boundaries, along with pressure and gravitation, contribute to the overall torque balance. Previous research by Rogister & Valette (2009) suggests that indeed rotational and gravito-inertial modes are coupled, thus shifting the frequencies of the Chandler Wobble (CW), the Free Core Nutation (FCN) and the Free Inner Core Nutation (FICN). Here we present the first results from a numerical model of the Earth's fluid core and its interaction with the rotational eigenmodes. In this first step we consider a fluid core without a solid inner core and we restrict to ellipticities of the same order as the Ekman number. We formulate the problem as a generalised eigenvalue problem that solves simultaneously the Liouville equation for the rotational modes (the torque balance), and the Navier-Stokes equation for the inertial modes.

Variational approach to stability boundary for the Taylor-Goldstein equation

NASA Astrophysics Data System (ADS)

Hirota, Makoto; Morrison, Philip J.

2015-11-01

Linear stability of inviscid stratified shear flow is studied by developing an efficient method for finding neutral (i.e., marginally stable) solutions of the Taylor-Goldstein equation. The classical Miles-Howard criterion states that stratified shear flow is stable if the local Richardson number JR is greater than 1/4 everywhere. In this work, the case of JR > 0 everywhere is considered by assuming strictly monotonic and smooth profiles of the ambient shear flow and density. It is shown that singular neutral modes that are embedded in the continuous spectrum can be found by solving one-parameter families of self-adjoint eigenvalue problems. The unstable ranges of wavenumber are searched for accurately and efficiently by adopting this method in a numerical algorithm. Because the problems are self-adjoint, the variational method can be applied to ascertain the existence of singular neutral modes. For certain shear flow and density profiles, linear stability can be proven by showing the non-existence of a singular neutral mode. New sufficient conditions, extensions of the Rayleigh-Fjortoft stability criterion for unstratified shear flows, are derived in this manner. This work was supported by JSPS Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation # 55053270.

SO(4) algebraic approach to the three-body bound state problem in two dimensions

NASA Astrophysics Data System (ADS)

Dmitrašinović, V.; Salom, Igor

2014-08-01

We use the permutation symmetric hyperspherical three-body variables to cast the non-relativistic three-body Schrödinger equation in two dimensions into a set of (possibly decoupled) differential equations that define an eigenvalue problem for the hyper-radial wave function depending on an SO(4) hyper-angular matrix element. We express this hyper-angular matrix element in terms of SO(3) group Clebsch-Gordan coefficients and use the latter's properties to derive selection rules for potentials with different dynamical/permutation symmetries. Three-body potentials acting on three identical particles may have different dynamical symmetries, in order of increasing symmetry, as follows: (1) S3 ⊗ OL(2), the permutation times rotational symmetry, that holds in sums of pairwise potentials, (2) O(2) ⊗ OL(2), the so-called "kinematic rotations" or "democracy symmetry" times rotational symmetry, that holds in area-dependent potentials, and (3) O(4) dynamical hyper-angular symmetry, that holds in hyper-radial three-body potentials. We show how the different residual dynamical symmetries of the non-relativistic three-body Hamiltonian lead to different degeneracies of certain states within O(4) multiplets.

Finite Temperature Densities via the S-Function Method with Application to Electron Screening in Plasmas

NASA Astrophysics Data System (ADS)

Watrous, Mitchell James

1997-12-01

A new approach to the Green's-function method for the calculation of equilibrium densities within the finite temperature, Kohn-Sham formulation of density functional theory is presented, which extends the method to all temperatures. The contour of integration in the complex energy plane is chosen such that the density is given by a sum of Green's function differences evaluated at the Matsubara frequencies, rather than by the calculation and summation of Kohn-Sham single-particle wave functions. The Green's functions are written in terms of their spectral representation and are calculated as the solutions of their defining differential equations. These differential equations are boundary value problems as opposed to the standard eigenvalue problems. For large values of the complex energy, the differential equations are further simplified from second to first-order by writing the Green's functions in terms of logarithmic derivatives. An asymptotic expression for the Green's functions is derived, which allows the sum over Matsubara poles to be approximated. The method is applied to the screening of nuclei by electrons in finite temperature plasmas. To demonstrate the method's utility, and to illustrate its advantages, the results of previous wave function type calculations for protons and neon nuclei are reproduced. The method is also used to formulate a new screening model for fusion reactions in the solar core, and the predicted reaction rate enhancements factors are compared with existing models.

The Pauli Objection

NASA Astrophysics Data System (ADS)

Leon, Juan; Maccone, Lorenzo

2017-12-01

Schrödinger's equation says that the Hamiltonian is the generator of time translations. This seems to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian. Then both time and energy must have the same spectrum since conjugate operators are unitarily equivalent. Clearly this is not always true: normal Hamiltonians have lower bounded spectrum and often only have discrete eigenvalues, whereas we typically desire that time can take any real value. Pauli concluded that constructing a general a time operator is impossible (although clearly it can be done in specific cases). Here we show how the Pauli argument fails when one uses an external system (a "clock") to track time, so that time arises as correlations between the system and the clock (conditional probability amplitudes framework). In this case, the time operator is conjugate to the clock Hamiltonian and not to the system Hamiltonian, but its eigenvalues still satisfy the Schrödinger equation for arbitrary system Hamiltonians.

On the identification of continuous vibrating systems modelled by hyperbolic partial differential equations

NASA Technical Reports Server (NTRS)

Udwadia, F. E.; Garba, J. A.

1983-01-01

This paper deals with the identification of spatially varying parameters in systems of finite spatial extent which can be described by second order hyperbolic differential equations. Two questions have been addressed. The first deals with 'partial identification' and inquires into the possibility of retrieving all the eigenvalues of the system from response data obtained at one location x-asterisk epsilon (0, 1). The second deals with the identification of the distributed coefficients rho(x), a(x) and b(x). Sufficient conditions for unique identification of all the eigenvalues of the system are obtained, and conditions under which the coefficients can be uniquely identified using suitable response data obtained at one point in the spatial domain are determined. Application of the results and their usefulness is demonstrated in the identification of the properties of tall building structural systems subjected to dynamic load environments.

Eigenvalue equation and core-mode cutoff of weakly guiding tapered fiber as three layer optical waveguide and used as biochemical sensor.

PubMed

Linslal, C L; Mohan, P M S; Halder, A; Gangopadhyay, T K

2012-06-01

The core-mode cutoff plays a major role in evanescent field absorption based sensors. A method has been proposed to calculate the core-mode cutoff by solving the eigenvalue equations of a weakly guiding three layer optical waveguide graphically. The variation of normalized waveguide parameter (V) is also calculated with different wavelengths at core-mode cutoff. At the first step, theoretical analysis of tapered fiber parameters has been performed for core-mode cutoff. The taper angle of an adiabatic tapered fiber is also analyzed using the length-scale criterion. Secondly, single-mode tapered fiber has been developed to make a precision sensor element suitable for chemical detection. Finally, the sensor element has been used to detect absorption peak of ethylenediamine. Results are presented in which an absorption peak at 1540 nm is observed.

A comparative study of history-based versus vectorized Monte Carlo methods in the GPU/CUDA environment for a simple neutron eigenvalue problem

NASA Astrophysics Data System (ADS)

Liu, Tianyu; Du, Xining; Ji, Wei; Xu, X. George; Brown, Forrest B.

2014-06-01

For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA's Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed.

A low dimensional dynamical system for the wall layer

NASA Technical Reports Server (NTRS)

Aubry, N.; Keefe, L. R.

1987-01-01

Low dimensional dynamical systems which model a fully developed turbulent wall layer were derived.The model is based on the optimally fast convergent proper orthogonal decomposition, or Karhunen-Loeve expansion. This decomposition provides a set of eigenfunctions which are derived from the autocorrelation tensor at zero time lag. Via Galerkin projection, low dimensional sets of ordinary differential equations in time, for the coefficients of the expansion, were derived from the Navier-Stokes equations. The energy loss to the unresolved modes was modeled by an eddy viscosity representation, analogous to Heisenberg's spectral model. A set of eigenfunctions and eigenvalues were obtained from direct numerical simulation of a plane channel at a Reynolds number of 6600, based on the mean centerline velocity and the channel width flow and compared with previous work done by Herzog. Using the new eigenvalues and eigenfunctions, a new ten dimensional set of ordinary differential equations were derived using five non-zero cross-stream Fourier modes with a periodic length of 377 wall units. The dynamical system was integrated for a range of the eddy viscosity prameter alpha. This work is encouraging.

Solutions of the Dirac Equation with the Shifted DENG-FAN Potential Including Yukawa-Like Tensor Interaction

NASA Astrophysics Data System (ADS)

Yahya, W. A.; Falaye, B. J.; Oluwadare, O. J.; Oyewumi, K. J.

2013-08-01

By using the Nikiforov-Uvarov method, we give the approximate analytical solutions of the Dirac equation with the shifted Deng-Fan potential including the Yukawa-like tensor interaction under the spin and pseudospin symmetry conditions. After using an improved approximation scheme, we solved the resulting schr\\"{o}dinger-like equation analytically. Numerical results of the energy eigenvalues are also obtained, as expected, the tensor interaction removes degeneracies between spin and pseudospin doublets.

Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire.

PubMed

Bacaër, Nicolas

2017-07-01

An explicit formula is found for the rate of extinction of subcritical linear birth-and-death processes in a random environment. The formula is illustrated by numerical computations of the eigenvalue with largest real part of the truncated matrix for the master equation. The generating function of the corresponding eigenvector satisfies a Fuchsian system of singular differential equations. A particular attention is set on the case of two environments, which leads to Riemann's differential equation.

Exact solution to the Schrödinger’s equation with pseudo-Gaussian potential

DOE Office of Scientific and Technical Information (OSTI.GOV)

Iacob, Felix, E-mail: felix@physics.uvt.ro; Lute, Marina, E-mail: marina.lute@upt.ro

2015-12-15

We consider the radial Schrödinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.

A semi-inverse variational method for generating the bound state energy eigenvalues in a quantum system: the Dirac Coulomb type-equation

NASA Astrophysics Data System (ADS)

Libarir, K.; Zerarka, A.

2018-05-01

Exact eigenspectra and eigenfunctions of the Dirac quantum equation are established using the semi-inverse variational method. This method improves of a considerable manner the efficiency and accuracy of results compared with the other usual methods much argued in the literature. Some applications for different state configurations are proposed to concretize the method.

A uniform object-oriented solution to the eigenvalue problem for real symmetric and Hermitian matrices

NASA Astrophysics Data System (ADS)

Castro, María Eugenia; Díaz, Javier; Muñoz-Caro, Camelia; Niño, Alfonso

2011-09-01

We present a system of classes, SHMatrix, to deal in a unified way with the computation of eigenvalues and eigenvectors in real symmetric and Hermitian matrices. Thus, two descendant classes, one for the real symmetric and other for the Hermitian cases, override the abstract methods defined in a base class. The use of the inheritance relationship and polymorphism allows handling objects of any descendant class using a single reference of the base class. The system of classes is intended to be the core element of more sophisticated methods to deal with large eigenvalue problems, as those arising in the variational treatment of realistic quantum mechanical problems. The present system of classes allows computing a subset of all the possible eigenvalues and, optionally, the corresponding eigenvectors. Comparison with well established solutions for analogous eigenvalue problems, as those included in LAPACK, shows that the present solution is competitive against them. Program summaryProgram title: SHMatrix Catalogue identifier: AEHZ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHZ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2616 No. of bytes in distributed program, including test data, etc.: 127 312 Distribution format: tar.gz Programming language: Standard ANSI C++. Computer: PCs and workstations. Operating system: Linux, Windows. Classification: 4.8. Nature of problem: The treatment of problems involving eigensystems is a central topic in the quantum mechanical field. Here, the use of the variational approach leads to the computation of eigenvalues and eigenvectors of real symmetric and Hermitian Hamiltonian matrices. Realistic models with several degrees of freedom leads to large (sometimes very large) matrices. Different techniques, such as divide and conquer, can be used to factorize the matrices in order to apply a parallel computing approach. However, it is still interesting to have a core procedure able to tackle the computation of eigenvalues and eigenvectors once the matrix has been factorized to pieces of enough small size. Several available software packages, such as LAPACK, tackled this problem under the traditional imperative programming paradigm. In order to ease the modelling of complex quantum mechanical models it could be interesting to apply an object-oriented approach to the treatment of the eigenproblem. This approach offers the advantage of a single, uniform treatment for the real symmetric and Hermitian cases. Solution method: To reach the above goals, we have developed a system of classes: SHMatrix. SHMatrix is composed by an abstract base class and two descendant classes, one for real symmetric matrices and the other for the Hermitian case. The object-oriented characteristics of inheritance and polymorphism allows handling both cases using a single reference of the base class. The basic computing strategy applied in SHMatrix allows computing subsets of eigenvalues and (optionally) eigenvectors. The tests performed show that SHMatrix is competitive, and more efficient for large matrices, than the equivalent routines of the LAPACK package. Running time: The examples included in the distribution take only a couple of seconds to run.

Modeling of nonequilibrium space plasma flows

NASA Technical Reports Server (NTRS)

Gombosi, Tamas

1995-01-01

Godunov-type numerical solution of the 20 moment plasma transport equations. One of the centerpieces of our proposal was the development of a higher order Godunov-type numerical scheme to solve the gyration dominated 20 moment transport equations. In the first step we explored some fundamental analytic properties of the 20 moment transport equations for a low b plasma, including the eigenvectors and eigenvalues of propagating disturbances. The eigenvalues correspond to wave speeds, while the eigenvectors characterize the transported physical quantities. In this paper we also explored the physically meaningful parameter range of the normalized heat flow components. In the second step a new Godunov scheme type numerical method was developed to solve the coupled set of 20 moment transport equations for a quasineutral single-ion plasma. The numerical method and the first results were presented at several national and international meetings and a paper describing the method has been published in the Journal of Computational Physics. To our knowledge this is the first numerical method which is capable of producing stable time-dependent solutions to the full 20 (or 16) moment set of transport equations, including the full heat flow equation. Previous attempts resulted in unstable (oscillating) solutions of the heat flow equations. Our group invested over two man-years into the development and implementation of the new method. The present model solves the 20 moment transport equations for an ion species and thermal electrons in 8 domain extending from a collision dominated to a collisionless region (200 km to 12,000 km). This model has been applied to study O+ acceleration due to Joule heating in the lower ionosphere.

Generalised quasiprobability distribution for Hermite polynomial squeezed states

NASA Astrophysics Data System (ADS)

Datta, Sunil; D'Souza, Richard

1996-02-01

Generalized quasiprobability distributions (QPD) for Hermite polynomial states are presented. These states are solutions of an eigenvalue equation which is quadratic in creation and annihilation operators. Analytical expressions for the QPD are presented for some special cases of the eigenvalues. For large squeezing these analytical expressions for the QPD take the form of a finite series in even Hermite functions. These expressions very transparently exhibit the transition between, P, Q and W functions corresponding to the change of the s-parameter of the QPD. Further, they clearly show the two-photon nature of the processes involved in the generation of these states.

Metal-coated magnetic nanoparticles in an optically active medium: A nonreciprocal metamaterial

NASA Astrophysics Data System (ADS)

Christofi, Aristi; Stefanou, Nikolaos

2018-03-01

We report on the optical response of a nonreciprocal bianisotropic metamaterial, consisting of spherical, metal-coated magnetic nanoparticles embedded in an optically active medium, thus combining gyrotropy, plasmonic resonances, and chirality in a versatile design. The corresponding effective medium is deduced by an appropriate two-step generalized Maxwell-Garnett homogenization scheme. The associated photonic band structure and transmission spectra are obtained through a six-vector formulation of Maxwell equations, which provides an efficient framework for general bianisotropic structures going beyond existing approaches that involve cumbersome nonlinear eigenvalue problems. Our results, analyzed and discussed in the light of group theory, provide evidence that the proposed metamaterial exhibits some remarkable frequency-tunable properties, such as strong, plasmon-enhanced nonreciprocal polarization azimuth rotation and magnetochiral dichroism.

Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zheng, Bin; Chen, Luoping; Hu, Xiaozhe

2016-03-05

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigatemore » the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.« less

The inference of atmospheric ozone using satellite nadir measurements in the 1042/cm band

NASA Technical Reports Server (NTRS)

Russell, J. M., III; Drayson, S. R.

1973-01-01

A description and detailed analysis of a technique for inferring atmospheric ozone information from satellite nadir measurements in the 1042 cm band are presented. A method is formulated for computing the emission from the lower boundary under the satellite which circumvents the difficult analytical problems caused by the presence of atmospheric clouds and the watervapor continuum absorption. The inversion equations are expanded in terms of the eigenvectors and eigenvalues of a least-squares-solution matrix, and an analysis is performed to determine the information content of the radiance measurements. Under favorable conditions there are only two pieces of independent information available from the measurements: (1) the total ozone and (2) the altitude of the primary maximum in the ozone profile.

Reduction of Free Edge Peeling Stress of Laminated Composites Using Active Piezoelectric Layers

PubMed Central

Huang, Bin; Kim, Heung Soo

2014-01-01

An analytical approach is proposed in the reduction of free edge peeling stresses of laminated composites using active piezoelectric layers. The approach is the extended Kantorovich method which is an iterative method. Multiterms of trial function are employed and governing equations are derived by taking the principle of complementary virtual work. The solutions are obtained by solving a generalized eigenvalue problem. By this approach, the stresses automatically satisfy not only the traction-free boundary conditions, but also the free edge boundary conditions. Through the iteration processes, the free edge stresses converge very quickly. It is found that the peeling stresses generated by mechanical loadings are significantly reduced by applying a proper electric field to the piezoelectric actuators. PMID:25025088

Quantum statistics of Raman scattering model with Stokes mode generation

NASA Technical Reports Server (NTRS)

Tanatar, Bilal; Shumovsky, Alexander S.

1994-01-01

The model describing three coupled quantum oscillators with decay of Rayleigh mode into the Stokes and vibration (phonon) modes is examined. Due to the Manley-Rowe relations the problem of exact eigenvalues and eigenstates is reduced to the calculation of new orthogonal polynomials defined both by the difference and differential equations. The quantum statistical properties are examined in the case when initially: the Stokes mode is in the vacuum state; the Rayleigh mode is in the number state; and the vibration mode is in the number of or squeezed states. The collapses and revivals are obtained for different initial conditions as well as the change in time the sub-Poisson distribution by the super-Poisson distribution and vice versa.

On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices

NASA Technical Reports Server (NTRS)

Fischer, Bernd; Freund, Roland W.

1992-01-01

The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.

Variational Methods For Sloshing Problems With Surface Tension

NASA Astrophysics Data System (ADS)

Tan, Chee Han; Carlson, Max; Hohenegger, Christel; Osting, Braxton

2016-11-01

We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in a container, including effects due to surface tension on the free surface. We restrict ourselves to a constant contact angle and we seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed to the zero surface tension case, where the problem reduces to a partial differential equation for the velocity potential, we obtain a coupled system for the velocity potential and the free surface displacement. We derive a new variational formulation of the coupled problem and establish the existence of solutions using the direct method from the Calculus of Variations. In the limit of zero surface tension, we recover the variational formulation of the classical Steklov eigenvalue problem, as derived by B. A. Troesch. For the particular case of an axially symmetric container, we propose a finite element numerical method for computing the sloshing modes of the coupled system. The scheme is implemented in FEniCS and we obtain a qualitative description of the effect of surface tension on the sloshing modes.

The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem

NASA Technical Reports Server (NTRS)

Jones, Mark T.; Patrick, Merrell L.

1989-01-01

The generalized eigenvalue problem, Kx = Lambda Mx, is of significant practical importance, especially in structural enginering where it arises as the vibration and buckling problem. A new algorithm, LANZ, based on Lanczos's method is developed. LANZ uses a technique called dynamic shifting to improve the efficiency and reliability of the Lanczos algorithm. A new algorithm for solving the tridiagonal matrices that arise when using Lanczos's method is described. A modification of Parlett and Scott's selective orthogonalization algorithm is proposed. Results from an implementation of LANZ on a Convex C-220 show it to be superior to a subspace iteration code.

Computer Solution of the Schrodinger Equation--Two Useful Programs.

ERIC Educational Resources Information Center

Evans, D. E.

1980-01-01

Describes a general purpose algorithm which enables one to calculate the allowed energy eigenvalues for an arbitrary potential. Results of a calculation where a centrifugal potential is added to the hydrogenic Coulomb potential are discussed. (Author/HM)

Determination of eigenvalues of dynamical systems by symbolic computation

NASA Technical Reports Server (NTRS)

Howard, J. C.

1982-01-01

A symbolic computation technique for determining the eigenvalues of dynamical systems is described wherein algebraic operations, symbolic differentiation, matrix formulation and inversion, etc., can be performed on a digital computer equipped with a formula-manipulation compiler. An example is included that demonstrates the facility with which the system dynamics matrix and the control distribution matrix from the state space formulation of the equations of motion can be processed to obtain eigenvalue loci as a function of a system parameter. The example chosen to demonstrate the technique is a fourth-order system representing the longitudinal response of a DC 8 aircraft to elevator inputs. This simplified system has two dominant modes, one of which is lightly damped and the other well damped. The loci may be used to determine the value of the controlling parameter that satisfied design requirements. The results were obtained using the MACSYMA symbolic manipulation system.

Fast Eigensolver for Computing 3D Earth's Normal Modes

NASA Astrophysics Data System (ADS)

Shi, J.; De Hoop, M. V.; Li, R.; Xi, Y.; Saad, Y.

2017-12-01

We present a novel parallel computational approach to compute Earth's normal modes. We discretize Earth via an unstructured tetrahedral mesh and apply the continuous Galerkin finite element method to the elasto-gravitational system. To resolve the eigenvalue pollution issue, following the analysis separating the seismic point spectrum, we utilize explicitly a representation of the displacement for describing the oscillations of the non-seismic modes in the fluid outer core. Effectively, we separate out the essential spectrum which is naturally related to the Brunt-Väisälä frequency. We introduce two Lanczos approaches with polynomial and rational filtering for solving this generalized eigenvalue problem in prescribed intervals. The polynomial filtering technique only accesses the matrix pair through matrix-vector products and is an ideal candidate for solving three-dimensional large-scale eigenvalue problems. The matrix-free scheme allows us to deal with fluid separation and self-gravitation in an efficient way, while the standard shift-and-invert method typically needs an explicit shifted matrix and its factorization. The rational filtering method converges much faster than the standard shift-and-invert procedure when computing all the eigenvalues inside an interval. Both two Lanczos approaches solve for the internal eigenvalues extremely accurately, comparing with the standard eigensolver. In our computational experiments, we compare our results with the radial earth model benchmark, and visualize the normal modes using vector plots to illustrate the properties of the displacements in different modes.

Periodic solutions for one dimensional wave equation with bounded nonlinearity

NASA Astrophysics Data System (ADS)

Ji, Shuguan

2018-05-01

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u (x) satisfies ess infηu (x) > 0 with ηu (x) = 1/2 u″/u - 1/4 (u‧/u)2, which actually excludes the classical constant coefficient model. For the case ηu (x) = 0, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form T = 2p-1/q (p , q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess infηu (x) > 0.

Incomplete augmented Lagrangian preconditioner for steady incompressible Navier-Stokes equations.

PubMed

Tan, Ning-Bo; Huang, Ting-Zhu; Hu, Ze-Jun

2013-01-01

An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids.

Linear stability analysis of the Vlasov-Poisson equations in high density plasmas in the presence of crossed fields and density gradients

NASA Technical Reports Server (NTRS)

Kaup, D. J.; Hansen, P. J.; Choudhury, S. Roy; Thomas, Gary E.

1986-01-01

The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.

KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

NASA Astrophysics Data System (ADS)

Chuluunbaatar, O.; Gusev, A. A.; Abrashkevich, A. G.; Amaya-Tapia, A.; Kaschiev, M. S.; Larsen, S. Y.; Vinitsky, S. I.

2007-10-01

A FORTRAN 77 program is presented which calculates energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach. In this approach, a multi-dimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on the finite interval with homogeneous boundary conditions of the third type. The resulting system of radial equations which contains the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite-element method. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials. Program summaryProgram title: KANTBP Catalogue identifier: ADZH_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZH_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4224 No. of bytes in distributed program, including test data, etc.: 31 232 Distribution format: tar.gz Programming language: FORTRAN 77 Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP RAM: depends on (a) the number of differential equations; (b) the number and order of finite-elements; (c) the number of hyperradial points; and (d) the number of eigensolutions required. Test run requires 30 MB Classification: 2.1, 2.4 External routines: GAULEG and GAUSSJ [W.H. Press, B.F. Flanery, S.A. Teukolsky, W.T. Vetterley, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986] Nature of problem: In the hyperspherical adiabatic approach [J. Macek, J. Phys. B 1 (1968) 831-843; U. Fano, Rep. Progr. Phys. 46 (1983) 97-165; C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77-142], a multi-dimensional Schrödinger equation for a two-electron system [A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro, Comput. Phys. Comm. 90 (1995) 311-339] or a hydrogen atom in magnetic field [M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352] is reduced by separating the radial coordinate ρ from the angular variables to a system of second-order ordinary differential equations which contain potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present the finite-element method procedure based on the use of high-order accuracy approximations for calculating approximate eigensolutions for such systems of coupled differential equations. Solution method: The boundary problems for coupled differential equations are solved by the finite-element method using high-order accuracy approximations [A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40-64]. The generalized algebraic eigenvalue problem AF=EBF with respect to pair unknowns ( E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982]. The generalized algebraic eigenvalue problem (A-EB)F=λDF with respect to pair unknowns (λ,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the LDL factorization of symmetric matrix and back-substitution methods using the DECOMP and REDBAK programs, respectively [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982]. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials described in [Yu. A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330-361; O. Chuluunbaatar, A.A. Gusev, S.Y. Larsen, S.I. Vinitsky, J. Phys. A 35 (2002) L513-L525; N.P. Mehta, J.R. Shepard, Phys. Rev. A 72 (2005) 032728-1-11; O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243-269]. For this benchmark model the needed analytical expressions for the potential matrix elements and first-derivative coupling terms, their asymptotics and asymptotics of radial solutions of the boundary problems for coupled differential equations have been produced with help of a MAPLE computer algebra system. Restrictions: The computer memory requirements depend on: (a) the number of differential equations; (b) the number and order of finite-elements; (c) the total number of hyperradial points; and (d) the number of eigensolutions required. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write-Up and listing for details). The user must also supply subroutine POTCAL for evaluating potential matrix elements. The user should supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMSC (when solving the scattering problem) that evaluate the asymptotics of the radial wave functions at the right boundary point in case of a boundary condition of the third type, respectively. Running time: The running time depends critically upon: (a) the number of differential equations; (b) the number and order of finite-elements; (c) the total number of hyperradial points on interval [0,ρ]; and (d) the number of eigensolutions required. The test run which accompanies this paper took 28.48 s without calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.

Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems.

PubMed

Pellacci, Benedetta; Verzini, Gianmaria

2018-05-01

We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. Our main result concerns the optimization of such threshold with respect to the fractional order [Formula: see text], the case [Formula: see text] corresponding to the standard Neumann Laplacian: when the habitat is not too fragmented, the principal positive eigenvalue can not have local minima for [Formula: see text]. As a consequence, the best strategy for survival is either following the diffusion with [Formula: see text] (i.e. Brownian diffusion), or with the lowest possible s (i.e. diffusion allowing long jumps), depending on the size of the domain. In addition, we show that analogous results hold for the standard fractional Laplacian in [Formula: see text], in periodic environments.

A variational eigenvalue solver on a photonic quantum processor

PubMed Central

Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter; Yung, Man-Hong; Zhou, Xiao-Qi; Love, Peter J.; Aspuru-Guzik, Alán; O’Brien, Jeremy L.

2014-01-01

Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansätze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry—calculating the ground-state molecular energy for He–H+. The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future. PMID:25055053

Parallel and Portable Monte Carlo Particle Transport

NASA Astrophysics Data System (ADS)

Lee, S. R.; Cummings, J. C.; Nolen, S. D.; Keen, N. D.

1997-08-01

We have developed a multi-group, Monte Carlo neutron transport code in C++ using object-oriented methods and the Parallel Object-Oriented Methods and Applications (POOMA) class library. This transport code, called MC++, currently computes k and α eigenvalues of the neutron transport equation on a rectilinear computational mesh. It is portable to and runs in parallel on a wide variety of platforms, including MPPs, clustered SMPs, and individual workstations. It contains appropriate classes and abstractions for particle transport and, through the use of POOMA, for portable parallelism. Current capabilities are discussed, along with physics and performance results for several test problems on a variety of hardware, including all three Accelerated Strategic Computing Initiative (ASCI) platforms. Current parallel performance indicates the ability to compute α-eigenvalues in seconds or minutes rather than days or weeks. Current and future work on the implementation of a general transport physics framework (TPF) is also described. This TPF employs modern C++ programming techniques to provide simplified user interfaces, generic STL-style programming, and compile-time performance optimization. Physics capabilities of the TPF will be extended to include continuous energy treatments, implicit Monte Carlo algorithms, and a variety of convergence acceleration techniques such as importance combing.

Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE

NASA Astrophysics Data System (ADS)

Forrester, Peter J.; Trinh, Allan K.

2018-05-01

The neighbourhood of the largest eigenvalue λmax in the Gaussian unitary ensemble (GUE) and Laguerre unitary ensemble (LUE) is referred to as the soft edge. It is known that there exists a particular centring and scaling such that the distribution of λmax tends to a universal form, with an error term bounded by 1/N2/3. We take up the problem of computing the exact functional form of the leading error term in a large N asymptotic expansion for both the GUE and LUE—two versions of the LUE are considered, one with the parameter a fixed and the other with a proportional to N. Both settings in the LUE case allow for an interpretation in terms of the distribution of a particular weighted path length in a model involving exponential variables on a rectangular grid, as the grid size gets large. We give operator theoretic forms of the corrections, which are corollaries of knowledge of the first two terms in the large N expansion of the scaled kernel and are readily computed using a method due to Bornemann. We also give expressions in terms of the solutions of particular systems of coupled differential equations, which provide an alternative method of computation. Both characterisations are well suited to a thinned generalisation of the original ensemble, whereby each eigenvalue is deleted independently with probability (1 - ξ). In Sec. V, we investigate using simulation the question of whether upon an appropriate centring and scaling a wider class of complex Hermitian random matrix ensembles have their leading correction to the distribution of λmax proportional to 1/N2/3.

More insights into early brain development through statistical analyses of eigen-structural elements of diffusion tensor imaging using multivariate adaptive regression splines

PubMed Central

Chen, Yasheng; Zhu, Hongtu; An, Hongyu; Armao, Diane; Shen, Dinggang; Gilmore, John H.; Lin, Weili

2013-01-01

The aim of this study was to characterize the maturational changes of the three eigenvalues (λ1 ≥ λ2 ≥ λ3) of diffusion tensor imaging (DTI) during early postnatal life for more insights into early brain development. In order to overcome the limitations of using presumed growth trajectories for regression analysis, we employed Multivariate Adaptive Regression Splines (MARS) to derive data-driven growth trajectories for the three eigenvalues. We further employed Generalized Estimating Equations (GEE) to carry out statistical inferences on the growth trajectories obtained with MARS. With a total of 71 longitudinal datasets acquired from 29 healthy, full-term pediatric subjects, we found that the growth velocities of the three eigenvalues were highly correlated, but significantly different from each other. This paradox suggested the existence of mechanisms coordinating the maturations of the three eigenvalues even though different physiological origins may be responsible for their temporal evolutions. Furthermore, our results revealed the limitations of using the average of λ2 and λ3 as the radial diffusivity in interpreting DTI findings during early brain development because these two eigenvalues had significantly different growth velocities even in central white matter. In addition, based upon the three eigenvalues, we have documented the growth trajectory differences between central and peripheral white matter, between anterior and posterior limbs of internal capsule, and between inferior and superior longitudinal fasciculus. Taken together, we have demonstrated that more insights into early brain maturation can be gained through analyzing eigen-structural elements of DTI. PMID:23455648

The Green's function in a channel with a sound-absorbing cover in the case of a uniform flow

NASA Astrophysics Data System (ADS)

Sobolev, A. F.

2012-07-01

We study the modal structure of an acoustic field of a point source as function of channel wall admittance in the case of a two-dimensional channel. The characteristic equation for determining the eigen-values corresponding to the boundary problem is studied in the form of this equation's dependence on the admittance, which varies in the entire complex plane. All modes, without exception, existing in the channel and forming the source field are classified based on the obtained topography of the characteristic equation. The expressions that describe the amplitudes and spatial distribution of the hydrodynamic modes, attenuation rate (for stable modes), or increment (for unstable modes) were obtained as functions of the wall admittance and flow velocity. It is shown that in addition to the hydrodynamic unstable modes existing downstream from the source, hydrodynamic unstable modes exist upstream from the source at any admittance. They appear only when the admittance has an elastic character. It is shown that hydrodynamic modes are induced only in the case when the source is located close to the wall or on the wall. The amplitude of these modes decreases exponentially with distance from the wall.

Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

DOE PAGES

Slaybaugh, R. N.; Ramirez-Zweiger, M.; Pandya, Tara; ...

2018-02-20

In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MGmore » Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.« less

Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

DOE Office of Scientific and Technical Information (OSTI.GOV)

Slaybaugh, R. N.; Ramirez-Zweiger, M.; Pandya, Tara

In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MGmore » Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.« less

A robust multilevel simultaneous eigenvalue solver

NASA Technical Reports Server (NTRS)

Costiner, Sorin; Taasan, Shlomo

1993-01-01

Multilevel (ML) algorithms for eigenvalue problems are often faced with several types of difficulties such as: the mixing of approximated eigenvectors by the solution process, the approximation of incomplete clusters of eigenvectors, the poor representation of solution on coarse levels, and the existence of close or equal eigenvalues. Algorithms that do not treat appropriately these difficulties usually fail, or their performance degrades when facing them. These issues motivated the development of a robust adaptive ML algorithm which treats these difficulties, for the calculation of a few eigenvectors and their corresponding eigenvalues. The main techniques used in the new algorithm include: the adaptive completion and separation of the relevant clusters on different levels, the simultaneous treatment of solutions within each cluster, and the robustness tests which monitor the algorithm's efficiency and convergence. The eigenvectors' separation efficiency is based on a new ML projection technique generalizing the Rayleigh Ritz projection, combined with a technique, the backrotations. These separation techniques, when combined with an FMG formulation, in many cases lead to algorithms of O(qN) complexity, for q eigenvectors of size N on the finest level. Previously developed ML algorithms are less focused on the mentioned difficulties. Moreover, algorithms which employ fine level separation techniques are of O(q(sub 2)N) complexity and usually do not overcome all these difficulties. Computational examples are presented where Schrodinger type eigenvalue problems in 2-D and 3-D, having equal and closely clustered eigenvalues, are solved with the efficiency of the Poisson multigrid solver. A second order approximation is obtained in O(qN) work, where the total computational work is equivalent to only a few fine level relaxations per eigenvector.

Generalized thermoelastic interaction in an isotropic solid cylinder without energy dissipation

NASA Astrophysics Data System (ADS)

Alshaikh, Fatimah

2018-04-01

In this paper, we constructed the generalized thermoelastic equations of an isotropic solid cylinder. The formulation is applied in the context of Green and Naghdi theory of types II (without energy dissipation). The material of the cylinder is supposed to be homogeneous isotropic both mechanically and thermally. The governing equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. Numerical results for the temperature distribution, displacement and radial stress are represented graphically.

Dynamic stability and bifurcation analysis in fractional thermodynamics

NASA Astrophysics Data System (ADS)

Béda, Péter B.

2018-02-01

In mechanics, viscoelasticity was the first field of applications in studying geomaterials. Further possibilities arise in spatial non-locality. Non-local materials were already studied in the 1960s by several authors as a part of continuum mechanics and are still in focus of interest because of the rising importance of materials with internal micro- and nano-structure. When material instability gained more interest, non-local behavior appeared in a different aspect. The problem was concerned to numerical analysis, because then instability zones exhibited singular properties for local constitutive equations. In dynamic stability analysis, mathematical aspects of non-locality were studied by using the theory of dynamic systems. There the basic set of equations describing the behavior of continua was transformed to an abstract dynamic system consisting of differential operators acting on the perturbation field variables. Such functions should satisfy homogeneous boundary conditions and act as indicators of stability of a selected state of the body under consideration. Dynamic systems approach results in conditions for cases, when the differential operators have critical eigenvalues of zero real parts (dynamic stability or instability conditions). When the critical eigenvalues have non-trivial eigenspace, the way of loss of stability is classified as a typical (or generic) bifurcation. Our experiences show that material non-locality and the generic nature of bifurcation at instability are connected, and the basic functions of the non-trivial eigenspace can be used to determine internal length quantities of non-local mechanics. Fractional calculus is already successfully used in thermo-elasticity. In the paper, non-locality is introduced via fractional strain into the constitutive relations of various conventional types. Then, by defining dynamic systems, stability and bifurcation are studied for states of thermo-mechanical solids. Stability conditions and genericity conditions are presented for constitutive relations under consideration.

Microscopic multiphonon approach to spectroscopy in the neutron-rich oxygen region

NASA Astrophysics Data System (ADS)

De Gregorio, G.; Knapp, F.; Lo Iudice, N.; Veselý, P.

2018-03-01

Background: A fairly rich amount of experimental spectroscopic data have disclosed intriguing properties of the nuclei in the region of neutron rich oxygen isotopes up to the neutron dripline. They, therefore, represent a unique laboratory for studying the evolution of nuclear structure away from the stability line. Purpose: We intend to give an exhaustive microscopic description of low and high energy spectra, dipole response, weak, and electromagnetic properties of the even 22O and the odd 23O and 23F. Method: An equation of motion phonon method generates an orthonormal basis of correlated n -phonon states (n =0 ,1 ,2 ,⋯ ) built of constituent Tamm-Dancoff phonons. This basis is adopted to solve the full eigenvalue equations in even nuclei and to construct an orthonormal particle-core basis for the eigenvalue problem in odd nuclei. No approximations are involved and the Pauli principle is taken into full account. The method is adopted to perform self-consistent, parameter free, calculations using an optimized chiral nucleon-nucleon interaction in a space encompassing up to two-phonon basis states. Results: The computed spectra in 22O and 23O and the dipole cross section in 22O are in overall agreement with the experimental data. The calculation describes poorly the spectrum of 23F. Conclusions: The two-phonon configurations play a crucial role in the description of spectra and transitions. The large discrepancies concerning the spectra of 23F are ultimately traced back to the large separation between the Hartree-Fock levels belonging to different major shells. We suggest that a more compact single particle spectrum is needed and can be generated by a new chiral potential which includes explicitly the contribution of the three-body forces.

Fully Parallel MHD Stability Analysis Tool

NASA Astrophysics Data System (ADS)

Svidzinski, Vladimir; Galkin, Sergei; Kim, Jin-Soo; Liu, Yueqiang

2014-10-01

Progress on full parallelization of the plasma stability code MARS will be reported. MARS calculates eigenmodes in 2D axisymmetric toroidal equilibria in MHD-kinetic plasma models. It is a powerful tool for studying MHD and MHD-kinetic instabilities and it is widely used by fusion community. Parallel version of MARS is intended for simulations on local parallel clusters. It will be an efficient tool for simulation of MHD instabilities with low, intermediate and high toroidal mode numbers within both fluid and kinetic plasma models, already implemented in MARS. Parallelization of the code includes parallelization of the construction of the matrix for the eigenvalue problem and parallelization of the inverse iterations algorithm, implemented in MARS for the solution of the formulated eigenvalue problem. Construction of the matrix is parallelized by distributing the load among processors assigned to different magnetic surfaces. Parallelization of the solution of the eigenvalue problem is made by repeating steps of the present MARS algorithm using parallel libraries and procedures. Initial results of the code parallelization will be reported. Work is supported by the U.S. DOE SBIR program.

Linear quadratic regulators with eigenvalue placement in a specified region

NASA Technical Reports Server (NTRS)

Shieh, Leang S.; Dib, Hani M.; Ganesan, Sekar

1988-01-01

A linear optimal quadratic regulator is developed for optimally placing the closed-loop poles of multivariable continuous-time systems within the common region of an open sector, bounded by lines inclined at + or - pi/2k (k = 2 or 3) from the negative real axis with a sector angle of pi/2 or less, and the left-hand side of a line parallel to the imaginary axis in the complex s-plane. The design method is mainly based on the solution of a linear matrix Liapunov equation, and the resultant closed-loop system with its eigenvalues in the desired region is optimal with respect to a quadratic performance index.

Systematic Construction and Calculation of Electronic Properties of Fullerene Series Related by Rotational Symmetry: From Fullerenes to Bicapped Nanotubes.

PubMed

Dias, Jerry Ray

2016-06-09

The results herein demonstrate that the methods of circumscribing and the facile calculation of Hückel molecular orbital (HMO) eigenvalues by mirror-plane fragmentation have a broad application in the construction of carbon cluster series and the systematic study of trends in their electronic properties. In comparing open-ended nanotubes and their isomeric elongated fullerenes (bicapped nanotubes), we show that the former are more aromatic but the latter are more conjugated and that progressive elongation increases aromaticity and conjugation in both. Recursion equations that will allow one to obtain the eigenvalues to all 5-endcapped nanotubes are given.

Detecting, anticipating, and predicting critical transitions in spatially extended systems.

PubMed

Kwasniok, Frank

2018-03-01

A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system. As a further step, a genuinely nonstationary POP analysis is introduced. Here, the system matrix of the linear stochastic model is time-dependent, allowing for extrapolation and prediction of instabilities beyond the learning data window. The methods are demonstrated and explored using the one-dimensional Swift-Hohenberg equation as an example, focusing on the dynamics of stochastic fluctuations around the homogeneous stable state prior to the first bifurcation. The POP-based techniques are able to extract and track the least stable eigenvalues and eigenvectors of the system; the nonstationary POP analysis successfully predicts the timing of the first instability and the unstable mode well beyond the learning data window.

Detecting, anticipating, and predicting critical transitions in spatially extended systems

NASA Astrophysics Data System (ADS)

Kwasniok, Frank

2018-03-01

A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system. As a further step, a genuinely nonstationary POP analysis is introduced. Here, the system matrix of the linear stochastic model is time-dependent, allowing for extrapolation and prediction of instabilities beyond the learning data window. The methods are demonstrated and explored using the one-dimensional Swift-Hohenberg equation as an example, focusing on the dynamics of stochastic fluctuations around the homogeneous stable state prior to the first bifurcation. The POP-based techniques are able to extract and track the least stable eigenvalues and eigenvectors of the system; the nonstationary POP analysis successfully predicts the timing of the first instability and the unstable mode well beyond the learning data window.

POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field

NASA Astrophysics Data System (ADS)

Chuluunbaatar, O.; Gusev, A. A.; Gerdt, V. P.; Rostovtsev, V. A.; Vinitsky, S. I.; Abrashkevich, A. G.; Kaschiev, M. S.; Serov, V. V.

2008-02-01

A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multi-channel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented. Program summaryProgram title:POTHMF Catalogue identifier:AEAA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAA_v1_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:8123 No. of bytes in distributed program, including test data, etc.:131 396 Distribution format:tar.gz Programming language:FORTRAN 77 Computer:Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV Operating system:OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP RAM:Depends on the number of radial differential equations; the number and order of finite elements; the number of radial points. Test run requires 4 MB Classification:2.5 External routines:POTHMF uses some Lapack routines, copies of which are included in the distribution (see README file for details). Nature of problem:In the multi-channel adiabatic approach the Schrödinger equation for a hydrogen-like atom in a homogeneous magnetic field of strength γ ( γ=B/B, B≅2.35×10 T is a dimensionless parameter which determines the field strength B) is reduced by separating the radial coordinate, r, from the angular variables, (θ,φ), and using a basis of the angular oblate spheroidal functions [3] to a system of second-order ordinary differential equations which contain first-derivative coupling terms [4]. The purpose of this program is to calculate potential curves and matrix elements of radial coupling needed for calculating the low-lying bound and scattering states of hydrogen-like atoms in a homogeneous magnetic field of strength 0<γ⩽1000 within the adiabatic approach [5]. The program evaluates also asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem needed to extract from the R-matrix a required symmetric shortrange open-channel reaction matrix K [6] independent from matching point [7]. In addition, the program computes the dipole transition matrix elements in the length form between the basis functions that are needed for calculating the dipole transitions between the low-lying bound and scattering states and photoionization cross sections [8]. Solution method:The angular oblate spheroidal eigenvalue problem depending on the radial variable is solved using a series expansion in the Legendre polynomials [3]. The resulting tridiagonal symmetric algebraic eigenvalue problem for the evaluation of selected eigenvalues, i.e. the potential curves, is solved by the LDLT factorization using the DSTEVR program [2]. Derivatives of the eigenfunctions with respect to the radial variable which are contained in matrix elements of the coupled radial equations are obtained by solving the inhomogeneous algebraic equations. The corresponding algebraic problem is solved by using the LDLT factorization with the help of the DPTTRS program [2]. Asymptotics of the matrix elements at large values of radial variable are computed using a series expansion in the associated Laguerre polynomials [9]. The corresponding matching points between the numeric and asymptotic solutions are found automatically. These asymptotics are used for the evaluation of the asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem [7]. As a test desk, the program is applied to the calculation of the energy values of the ground and excited bound states and reaction matrix of multi-channel scattering problem for a hydrogen atom in a homogeneous magnetic field using the KANTBP program [10]. Restrictions:The computer memory requirements depend on: the number of radial differential equations; the number and order of finite elements; the total number of radial points. Restrictions due to dimension sizes can be changed by resetting a small number of PARAMETER statements before recompiling (see Introduction and listing for details). Running time:The running time depends critically upon: the number of radial differential equations; the number and order of finite elements; the total number of radial points on interval [r,r]. The test run which accompanies this paper took 7 s required for calculating of potential curves, radial matrix elements, and dipole transition matrix elements on a finite-element grid on interval [ r=0, r=100] used for solving discrete and continuous spectrum problems and obtaining asymptotic regular and irregular matrix radial solutions at r=100 for continuous spectrum problem on the Intel Pentium IV 2.4 GHz. The number of radial differential equations was equal to 6. The accompanying test run using the KANTBP program took 2 s for solving discrete and continuous spectrum problems using the above calculated potential curves, matrix elements and asymptotic regular and irregular matrix radial solutions. Note, that in the accompanied benchmark calculations of the photoionization cross-sections from the bound states of a hydrogen atom in a homogeneous magnetic field to continuum we have used interval [ r=0, r=1000] for continuous spectrum problem. The total number of radial differential equations was varied from 10 to 18. References:W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986. http://www.netlib.org/lapack/. M. Abramovits, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127; A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640; C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (Eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320; U. Fano, A.R.P. Rau, Atomic Collisions and Spectra, Academic Press, Florida, 1986. M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352; O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, V.V. Serov, T.V. Tupikova, S.I. Vinitsky, Proc. SPIE 6537 (2007) 653706-1-18. M.J. Seaton, Rep. Prog. Phys. 46 (1983) 167-257. M. Gailitis, J. Phys. B 9 (1976) 843-854; J. Macek, Phys. Rev. A 30 (1984) 1277-1278; S.I. Vinitsky, V.P. Gerdt, A.A. Gusev, M.S. Kaschiev, V.A. Rostovtsev, V.N. Samoylov, T.V. Tupikova, O. Chuluunbaatar, Programming and Computer Software 33 (2007) 105-116. H. Friedrich, Theoretical Atomic Physics, Springer, New York, 1991. R.J. Damburg, R.Kh. Propin, J. Phys. B 1 (1968) 681-691; J.D. Power, Phil. Trans. Roy. Soc. London A 274 (1973) 663-702. O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649-675.

Incomplete Augmented Lagrangian Preconditioner for Steady Incompressible Navier-Stokes Equations

PubMed Central

Tan, Ning-Bo; Huang, Ting-Zhu; Hu, Ze-Jun

2013-01-01

An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids. PMID:24235888

Observability of discretized partial differential equations

NASA Technical Reports Server (NTRS)

Cohn, Stephen E.; Dee, Dick P.

1988-01-01

It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.

Qualitative properties of large buckled states of spherical shells

NASA Technical Reports Server (NTRS)

Shih, K. G.; Antman, S. S.

1985-01-01

A system of 6th-order quasi-linear Ordinary Differential Equations is analyzed to show the global existence of axisymmetrically buckled states. A surprising nodal property is obtained which shows that everywhere along a branch of solutions that bifurcates from a simple eigenvalue of the linearized equation, the number of simultaneously vanishing points of both shear resultant and circumferential bending moment resultant remains invariant, provided that a certain auxiliary condition is satisfied.

AKNS eigenvalue spectrum for densely spaced envelope solitary waves

NASA Astrophysics Data System (ADS)

Slunyaev, Alexey; Starobor, Alexey

2010-05-01

The problem of the influence of one envelope soliton to the discrete eigenvalues of the associated scattering problem for the other envelope soliton, which is situated close to the first one, is discussed. Envelope solitons are exact solutions of the integrable nonlinear Schrödinger equation (NLS). Their generalizations (taking into account the background nonlinear waves [1-4] or strongly nonlinear effects [5, 6]) are possible candidates to rogue waves in the ocean. The envelope solitary waves could be in principle detected in the stochastic wave field by approaches based on the Inverse Scattering Technique in terms of ‘unstable modes' (see [1-3]), or envelope solitons [7-8]. However, densely spaced intense groups influence the spectrum of the associated scattering problem, so that the solitary trains cannot be considered alone. Here we solve the initial-value problem exactly for some simplified configurations of the wave field, representing two closely placed intense wave groups, within the frameworks of the NLS equation by virtue of the solution of the AKNS system [9]. We show that the analogues of the level splitting and the tunneling effects, known in quantum physics, exist in the context of the NLS equation, and thus may be observed in application to sea waves [10]. These effects make the detecting of single solitary wave groups surrounded by other nonlinear wave groups difficult. [1]. A.L. Islas, C.M. Schober (2005) Predicting rogue waves in random oceanic sea states. Phys. Fluids 17, 031701-1-4. [2]. A.R. Osborne, M. Onorato, M. Serio (2005) Nonlinear Fourier analysis of deep-water random surface waves: Theoretical formulation and and experimental observations of rogue waves. 14th Aha Huliko's Winter Workshop, Honolulu, Hawaii. [3]. C.M. Schober, A. Calini (2008) Rogue waves in higher order nonlinear Schrödinger models. In: Extreme Waves (Eds.: E. Pelinovsky & C. Kharif), Springer. [4]. N. Akhmediev, A. Ankiewicz, M. Taki (2009) Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675-678. [5]. A.I. Dyachenko, V.E. Zakharov (2008) On the formation of freak waves on the surface of deep water. JETP Lett. 88 (5), 307-311. [6]. A.V. Slunyaev (2009) Numerical simulation of "limiting" envelope solitons of gravity waves on deep water. JETP 109, 676-686. [7]. A. Slunyaev, E. Pelinovsky, and C. Guedes Soares (2005) Modeling freak waves from the North Sea. Appl. Ocean Res. 27, 12-22. [8]. A. Slunyaev (2006) Nonlinear analysis and simulations of measured freak wave time series. Eur. J. Mech. B / Fluids 25, 621-635. [9]. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur (1974) The inverse scattering transform - Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249-315. [10]. A.V. Starobor (2009) Interpretation of the inverse scattering data for the analysis of wave groups on water surface. Bachelor degree thesis. N. Novgorod State University, in Russian.

Topics in Bethe Ansatz

NASA Astrophysics Data System (ADS)

Wang, Chunguang

Integrable quantum spin chains have close connections to integrable quantum field. theories, modern condensed matter physics, string and Yang-Mills theories. Bethe. ansatz is one of the most important approaches for solving quantum integrable spin. chains. At the heart of the algebraic structure of integrable quantum spin chains is. the quantum Yang-Baxter equation and the boundary Yang-Baxter equation. This. thesis focuses on four topics in Bethe ansatz. The Bethe equations for the isotropic periodic spin-1/2 Heisenberg chain with N. sites have solutions containing ±i/2 that are singular: both the corresponding energy and the algebraic Bethe ansatz vector are divergent. Such solutions must be carefully regularized. We consider a regularization involving a parameter that can be. determined using a generalization of the Bethe equations. These generalized Bethe. equations provide a practical way of determining which singular solutions correspond. to eigenvectors of the model. The Bethe equations for the periodic XXX and XXZ spin chains admit singular. solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to derive conditions for such singular solutions to bephysical, in which case they correspond to genuine eigenvalues and eigenvectors of. the Hamiltonian. We analyze the ground state of the open spin-1/2 isotropic quantum spin chain. with a non-diagonal boundary term using a recently proposed Bethe ansatz solution. As the coefficient of the non-diagonal boundary term tends to zero, the Bethe roots. split evenly into two sets: those that remain finite, and those that become infinite. We. argue that the former satisfy conventional Bethe equations, while the latter satisfy a. generalization of the Richardson-Gaudin equations. We derive an expression for the. leading correction to the boundary energy in terms of the boundary parameters. We argue that the Hamiltonians for A(2) 2n open quantum spin chains corresponding. to two choices of integrable boundary conditions have the symmetries Uq(Bn) and. Uq(Cn), respectively. The deformation of Cn is novel, with a nonstandard coproduct. We find a formula for the Dynkin labels of the Bethe states (which determine the degeneracies of the corresponding eigenvalues) in terms of the numbers of Bethe roots of. each type. With the help of this formula, we verify numerically (for a generic value of. the anisotropy parameter) that the degeneracies and multiplicities of the spectra implied by the quantum group symmetries are completely described by the Bethe ansatz.

Three-body spectrum in a finite volume: The role of cubic symmetry

DOE PAGES

Doring, M.; Hammer, H. -W.; Mai, M.; ...

2018-06-15

The three-particle quantization condition is partially diagonalized in the center-of-mass frame by using cubic symmetry on the lattice. To this end, instead of spherical harmonics, the kernel of the Bethe-Salpeter equation for particle-dimer scattering is expanded in the basis functions of different irreducible representations of the octahedral group. Such a projection is of particular importance for the three-body problem in the finite volume due to the occurrence of three-body singularities above breakup. Additionally, we study the numerical solution and properties of such a projected quantization condition in a simple model. It is shown that, for large volumes, these solutions allowmore » for an instructive interpretation of the energy eigenvalues in terms of bound and scattering states.« less

Three-body spectrum in a finite volume: The role of cubic symmetry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Doring, M.; Hammer, H. -W.; Mai, M.

The three-particle quantization condition is partially diagonalized in the center-of-mass frame by using cubic symmetry on the lattice. To this end, instead of spherical harmonics, the kernel of the Bethe-Salpeter equation for particle-dimer scattering is expanded in the basis functions of different irreducible representations of the octahedral group. Such a projection is of particular importance for the three-body problem in the finite volume due to the occurrence of three-body singularities above breakup. Additionally, we study the numerical solution and properties of such a projected quantization condition in a simple model. It is shown that, for large volumes, these solutions allowmore » for an instructive interpretation of the energy eigenvalues in terms of bound and scattering states.« less

Three-dimensional computation of laser cavity eigenmodes by the use of finite element analysis (FEA)

NASA Astrophysics Data System (ADS)

Altmann, Konrad; Pflaum, Christoph; Seider, David

2004-06-01

A new method for computing eigenmodes of a laser resonator by the use of finite element analysis (FEA) is presented. For this purpose, the scalar wave equation [Δ + k2]E(x,y,z) = 0 is transformed into a solvable 3D eigenvalue problem by separating out the propagation factor exp(-ikz) from the phasor amplitude E(x,y,z) of the time-harmonic electrical field. For standing wave resonators, the beam inside the cavity is represented by a two-wave ansatz. For cavities with parabolic optical elements the new approach has successfully been verified by the use of the Gaussian mode algorithm. For a DPSSL with a thermally lensing crystal inside the cavity the expected deviation between Gaussian approximation and numerical solution could be demonstrated clearly.

Propagation Constant of a Rectangular Waveguides Completely Full of Ferrite Magnetized Longitudinally

NASA Astrophysics Data System (ADS)

Sakli, Hedi; Benzina, Hafedh; Aguili, Taoufik; Tao, Jun Wu

2009-08-01

This paper is an analysis of rectangular waveguide completely full of ferrite magnetized longitudinally. The analysis is based on the formulation of the transverse operator method (TOM), followed by the application of the Galerkin method. We obtain an eigenvalue equation system. The propagation constant of some homogenous and anisotropic waveguide structures with ferrite has been obtained. The results presented here show that the transverse operator formulation is not only an elegant theoretical form, but also a powerful and efficient analysis method, it is useful to solve a number of the propagation problems in electromagnetic. One advantage of this method is that it presents a fast convergence. Numerical examples are given for different cases and compared with the published results. A good agreement is obtained.

Swimming of a sphere in a viscous incompressible fluid with inertia

NASA Astrophysics Data System (ADS)

Felderhof, B. U.; Jones, R. B.

2017-08-01

The swimming of a sphere immersed in a viscous incompressible fluid with inertia is studied for surface modulations of small amplitude on the basis of the Navier-Stokes equations. The mean swimming velocity and the mean rate of dissipation are expressed as quadratic forms in term of the surface displacements. With a choice of a basis set of modes the quadratic forms correspond to two Hermitian matrices. Optimization of the mean swimming velocity for given rate of dissipation requires the solution of a generalized eigenvalue problem involving the two matrices. It is found for surface modulations of low multipole order that the optimal swimming efficiency depends in intricate fashion on a dimensionless scale number involving the radius of the sphere, the period of the cycle, and the kinematic viscosity of the fluid.

Non-algebraic integrability of the Chew-Low reversible dynamical system of the Cremona type and the relation with the 7th Hilbert problem (non-resonant case)

NASA Astrophysics Data System (ADS)

Rerikh, K. V.

A smooth reversible dynamical system (SRDS) and a system of nonlinear functional equations, defined by a certain rational quadratic Cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions (the Chew-Low equations for p- wave πN- scattering) are considered. This SRDS is splitted into 1- and 2-dimensional ones. An explicit Cremona transformation that completely determines the exact solution of the two-dimensional system is found. This solution depends on an odd function satisfying a nonlinear autonomous 3-point functional equation. Non-algebraic integrability of SRDS under consideration is proved using the method of Poincaré normal forms and the Siegel theorem on biholomorphic linearization of a mapping at a non-resonant fixed point. The proof is based on the classical Feldman-Baker theorem on linear forms of logarithms of algebraic numbers, which, in turn, relies upon solving the 7th Hilbert problem by A.I. Gel'fond and T. Schneider and new powerful methods of A. Baker in the theory of transcendental numbers. The general theorem, following from the Feldman-Baker theorem, on applicability of the Siegel theorem to the set of the eigenvalues λ ɛ Cn of a mapping at a non-resonant fixed point which belong to the algebraic number field A is formulated and proved. The main results are presented in Theorems 1-3, 5, 7, 8 and Remarks 3, 7.

The three-body problem with short-range interactions

NASA Astrophysics Data System (ADS)

Nielsen, E.; Fedorov, D. V.; Jensen, A. S.; Garrido, E.

2001-06-01

The quantum mechanical three-body problem is studied for general short-range interactions. We work in coordinate space to facilitate accurate computations of weakly bound and spatially extended systems. Hyperspherical coordinates are used in both the interpretation and as an integral part of the numerical method. Universal properties and model independence are discussed throughout the report. We present an overview of the hyperspherical adiabatic Faddeev equations. The wave function is expanded on hyperspherical angular eigenfunctions which in turn are found numerically using the Faddeev equations. We generalize the formalism to any dimension of space d greater or equal to two. We present two numerical techniques for solving the Faddeev equations on the hypersphere. These techniques are effective for short and intermediate/large distances including use for hard core repulsive potentials. We study the asymptotic limit of large hyperradius and derive the analytic behaviour of the angular eigenvalues and eigenfunctions. We discuss four applications of the general method. We first analyze the Efimov and Thomas effects for arbitrary angular momenta and for arbitrary dimensions d. Second we apply the method to extract the general behaviour of weakly bound three-body systems in two dimensions. Third we illustrate the method in three dimensions by structure computations of Borromean halo nuclei, the hypertriton and helium molecules. Fourth we investigate in three dimensions three-body continuum properties of Borromean halo nuclei and recombination reactions of helium atoms as an example of direct relevance for the stability of Bose-Einstein condensates.

Analysis of the Pre-stack Split-Step Migration Operator Using Ritz Values

NASA Astrophysics Data System (ADS)

Kaplan, S. T.; Sacchi, M. D.

2009-05-01

The Born approximation for the acoustic wave-field is often used as a basis for developing algorithms in seismic imaging (migration). The approximation is linear, and, as such, can be written as a matrix-vector multiplication (Am=d). In the seismic imaging problem, d is seismic data (the recorded wave-field), and we aim to find the seismic reflectivity m (a representation of earth structure and properties) so that Am=d is satisfied. This is the often studied inverse problem of seismic migration, where given A and d, we solve for m. This can be done in a least-squares sense, so that the equation of interest is, AHAm = AHd. Hence, the solution m is largely dependent on the properties of AHA. The imaging Jacobian J provides an approximation to AHA, so that J-1AHA is, in a broad sense, better behaved then AHA. We attempt to quantify this last statement by providing an analysis of AHA and J-1AHA using their Ritz values, and for the particular case where A is built using a pre-stack split-step migration algorithm. Typically, one might try to analyze the behaviour of these matrices using their eigenvalue spectra. The difficulty in the analysis of AHA and J-1AHA lie in their size. For example, a subset of the relatively small Marmousi data set makes AHA a complex valued matrix with, roughly, dimensions of 45 million by 45 million (requiring, in single-precision, about 16 Peta-bytes of computer memory). In short, the size of the matrix makes its eigenvalues difficult to compute. Instead, we compute the leading principal minors of similar tridiagonal matrices, Bk=Vk-1AHAVk and Ck = Uk-1 J-1 AHAUk. These can be constructed using, for example, the Lanczos decomposition. Up to some value of k it is feasible to compute the eigenvalues of Bk and Ck which, in turn, are the Ritz values of, respectively, AHA and J-1 AHA, and may allow us to make quantitative statements about their behaviours.

A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lerche, I.; Low, B. C.

2014-10-15

An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θB{sub φ}=Q(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin θB{sub φ}=aA|A|{sup 1/n}, n a positive constant, admits separable fields with A=(A{sub n}(θ))/(r{sup n}) , posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and A{sub n}(θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship betweenmore » the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B=(H(θ,φ))/(r{sup n+2}) promises field solutions of even richer topological varieties but allowing for φ-dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4/3 as discussed in the Appendix.« less

Solution of D dimensional Dirac equation for coulombic potential using NU method and its thermodynamics properties

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cari, C., E-mail: cari@staff.uns.ac.id; Suparmi, A., E-mail: soeparmi@staff.uns.ac.id; Yunianto, M., E-mail: muhtaryunianto@staff.uns.ac.id

2016-02-08

The analytical solution of Ddimensional Dirac equation for Coulombic potential is investigated using Nikiforov-Uvarov method. The D dimensional relativistic energy spectra are obtained from relativistic energy eigenvalue equation by using Mat Lab software.The corresponding D dimensional radial wave functions are formulated in the form of generalized Jacobi and Laguerre Polynomials. In the non-relativistic limit, the relativistic energy equation reduces to the non-relativistic energy which will be applied to determine some thermodynamical properties of the system. The thermodynamical properties of the system are expressed in terms of error function and imaginary error function.

Numerical solution of stiff systems of ordinary differential equations with applications to electronic circuits

NASA Technical Reports Server (NTRS)

Rosenbaum, J. S.

1971-01-01

Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

Monte Carlo Techniques for Nuclear Systems - Theory Lectures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brown, Forrest B.

These are lecture notes for a Monte Carlo class given at the University of New Mexico. The following topics are covered: course information; nuclear eng. review & MC; random numbers and sampling; computational geometry; collision physics; tallies and statistics; eigenvalue calculations I; eigenvalue calculations II; eigenvalue calculations III; variance reduction; parallel Monte Carlo; parameter studies; fission matrix and higher eigenmodes; doppler broadening; Monte Carlo depletion; HTGR modeling; coupled MC and T/H calculations; fission energy deposition. Solving particle transport problems with the Monte Carlo method is simple - just simulate the particle behavior. The devil is in the details, however. Thesemore » lectures provide a balanced approach to the theory and practice of Monte Carlo simulation codes. The first lectures provide an overview of Monte Carlo simulation methods, covering the transport equation, random sampling, computational geometry, collision physics, and statistics. The next lectures focus on the state-of-the-art in Monte Carlo criticality simulations, covering the theory of eigenvalue calculations, convergence analysis, dominance ratio calculations, bias in Keff and tallies, bias in uncertainties, a case study of a realistic calculation, and Wielandt acceleration techniques. The remaining lectures cover advanced topics, including HTGR modeling and stochastic geometry, temperature dependence, fission energy deposition, depletion calculations, parallel calculations, and parameter studies. This portion of the class focuses on using MCNP to perform criticality calculations for reactor physics and criticality safety applications. It is an intermediate level class, intended for those with at least some familiarity with MCNP. Class examples provide hands-on experience at running the code, plotting both geometry and results, and understanding the code output. The class includes lectures & hands-on computer use for a variety of Monte Carlo calculations. Beginning MCNP users are encouraged to review LA-UR-09-00380, "Criticality Calculations with MCNP: A Primer (3nd Edition)" (available at http:// mcnp.lanl.gov under "Reference Collection") prior to the class. No Monte Carlo class can be complete without having students write their own simple Monte Carlo routines for basic random sampling, use of the random number generator, and simplified particle transport simulation.« less

Computation of resistive instabilities by matched asymptotic expansions

DOE PAGES

Glasser, A. H.; Wang, Z. R.; Park, J. -K.

2016-11-17

Here, we present a method for determining the linear resistive magnetohydrodynamic (MHD) stability of an axisymmetric toroidal plasma, based on the method of matched asymptotic expansions. The plasma is partitioned into a set of ideal MHD outer regions, connected through resistive MHD inner regions about singular layers where q = m/n, with m and n toroidal mode numbers, respectively, and q the safety factor. The outer regions satisfy the ideal MHD equations with zero-frequency, which are identical to the Euler-Lagrange equations for minimizing the potential energy delta W. The solutions to these equations go to infinity at the singular surfaces.more » The inner regions satisfy the equations of motion of resistive MHD with a finite eigenvalue, resolving the singularity. Both outer and inner regions are solved numerically by newly developed singular Galerkin methods, using specialized basis functions. These solutions are matched asymptotically, providing a complex dispersion relation which is solved for global eigenvalues and eigenfunctions in full toroidal geometry. The dispersion relation may have multiple complex unstable roots, which are found by advanced root-finding methods. These methods are much faster and more robust than the previous numerical methods. The new methods are applicable to more challenging high-pressure and strongly shaped plasma equilibria and generalizable to more realistic inner region dynamics. In the thermonuclear regime, where the outer and inner regions overlap, they are also much faster and more accurate than the straight-through methods, which treat the resistive MHD equations in the whole plasma volume.« less

Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Besse, Nicolas, E-mail: Nicolas.Besse@oca.eu; Institut Jean Lamour, UMR CNRS/UL 7198, Université de Lorraine, BP 70239 54506 Vandoeuvre-lès-Nancy Cedex; Coulette, David, E-mail: David.Coulette@ipcms.unistra.fr

2016-08-15

Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov–Poisson and Vlasov–Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to themore » VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, “Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry” (submitted)] and were found to be surprisingly close to those for the original gyrokinetic-Vlasov equations. The purpose of the present paper is to make these new ideas accessible to two readerships: applied mathematicians and plasma physicists.« less

Introducing Computational Approaches in Intermediate Mechanics

NASA Astrophysics Data System (ADS)

Cook, David M.

2006-12-01

In the winter of 2003, we at Lawrence University moved Lagrangian mechanics and rigid body dynamics from a required sophomore course to an elective junior/senior course, freeing 40% of the time for computational approaches to ordinary differential equations (trajectory problems, the large amplitude pendulum, non-linear dynamics); evaluation of integrals (finding centers of mass and moment of inertia tensors, calculating gravitational potentials for various sources); and finding eigenvalues and eigenvectors of matrices (diagonalizing the moment of inertia tensor, finding principal axes), and to generating graphical displays of computed results. Further, students begin to use LaTeX to prepare some of their submitted problem solutions. Placed in the middle of the sophomore year, this course provides the background that permits faculty members as appropriate to assign computer-based exercises in subsequent courses. Further, students are encouraged to use our Computational Physics Laboratory on their own initiative whenever that use seems appropriate. (Curricular development supported in part by the W. M. Keck Foundation, the National Science Foundation, and Lawrence University.)

Interplay between parity-time symmetry, supersymmetry, and nonlinearity: An analytically tractable case example

DOE PAGES

Kevrekidis, Panayotis G.; Cuevas–Maraver, Jesús; Saxena, Avadh; ...

2015-10-01

In the present work, we combine the notion of parity-time (PT) symmetry with that of supersymmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the so-called Pöschl-Teller potential which is real. Not only are we able to identify and numerically confirm the eigenvalues of the relevant problem, but we also show that the corresponding nonlinear problem, in the presence of an arbitrary power-law nonlinearity, has an exact bright soliton solution that can be analytically identified and has intriguing stability properties, such as an oscillatory instability, which is absent for the corresponding solutionmore » of the regular nonlinear Schrödinger equation with arbitrary power-law nonlinearity. The spectral properties and dynamical implications of this instability are examined. Furthermore, we believe that these findings may pave the way toward initiating a fruitful interplay between the notions of PT symmetry, supersymmetric partner potentials, and nonlinear interactions.« less

Interplay between parity-time symmetry, supersymmetry, and nonlinearity: An analytically tractable case example

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kevrekidis, Panayotis G.; Cuevas–Maraver, Jesús; Saxena, Avadh

In the present work, we combine the notion of parity-time (PT) symmetry with that of supersymmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the so-called Pöschl-Teller potential which is real. Not only are we able to identify and numerically confirm the eigenvalues of the relevant problem, but we also show that the corresponding nonlinear problem, in the presence of an arbitrary power-law nonlinearity, has an exact bright soliton solution that can be analytically identified and has intriguing stability properties, such as an oscillatory instability, which is absent for the corresponding solutionmore » of the regular nonlinear Schrödinger equation with arbitrary power-law nonlinearity. The spectral properties and dynamical implications of this instability are examined. Furthermore, we believe that these findings may pave the way toward initiating a fruitful interplay between the notions of PT symmetry, supersymmetric partner potentials, and nonlinear interactions.« less

Optimal exponential synchronization of general chaotic delayed neural networks: an LMI approach.

PubMed

Liu, Meiqin

2009-09-01

This paper investigates the optimal exponential synchronization problem of general chaotic neural networks with or without time delays by virtue of Lyapunov-Krasovskii stability theory and the linear matrix inequality (LMI) technique. This general model, which is the interconnection of a linear delayed dynamic system and a bounded static nonlinear operator, covers several well-known neural networks, such as Hopfield neural networks, cellular neural networks (CNNs), bidirectional associative memory (BAM) networks, and recurrent multilayer perceptrons (RMLPs) with or without delays. Using the drive-response concept, time-delay feedback controllers are designed to synchronize two identical chaotic neural networks as quickly as possible. The control design equations are shown to be a generalized eigenvalue problem (GEVP) which can be easily solved by various convex optimization algorithms to determine the optimal control law and the optimal exponential synchronization rate. Detailed comparisons with existing results are made and numerical simulations are carried out to demonstrate the effectiveness of the established synchronization laws.

A Decentralized Eigenvalue Computation Method for Spectrum Sensing Based on Average Consensus

NASA Astrophysics Data System (ADS)

Mohammadi, Jafar; Limmer, Steffen; Stańczak, Sławomir

2016-07-01

This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.

Eigenvalue statistics for the sum of two complex Wishart matrices

NASA Astrophysics Data System (ADS)

Kumar, Santosh

2014-09-01

The sum of independent Wishart matrices, taken from distributions with unequal covariance matrices, plays a crucial role in multivariate statistics, and has applications in the fields of quantitative finance and telecommunication. However, analytical results concerning the corresponding eigenvalue statistics have remained unavailable, even for the sum of two Wishart matrices. This can be attributed to the complicated and rotationally noninvariant nature of the matrix distribution that makes extracting the information about eigenvalues a nontrivial task. Using a generalization of the Harish-Chandra-Itzykson-Zuber integral, we find exact solution to this problem for the complex Wishart case when one of the covariance matrices is proportional to the identity matrix, while the other is arbitrary. We derive exact and compact expressions for the joint probability density and marginal density of eigenvalues. The analytical results are compared with numerical simulations and we find perfect agreement.

Shape sensitivity analysis of flutter response of a laminated wing

NASA Technical Reports Server (NTRS)

Bergen, Fred D.; Kapania, Rakesh K.

1988-01-01

A method is presented for calculating the shape sensitivity of a wing aeroelastic response with respect to changes in geometric shape. Yates' modified strip method is used in conjunction with Giles' equivalent plate analysis to predict the flutter speed, frequency, and reduced frequency of the wing. Three methods are used to calculate the sensitivity of the eigenvalue. The first method is purely a finite difference calculation of the eigenvalue derivative directly from the solution of the flutter problem corresponding to the two different values of the shape parameters. The second method uses an analytic expression for the eigenvalue sensitivities of a general complex matrix, where the derivatives of the aerodynamic, mass, and stiffness matrices are computed using a finite difference approximation. The third method also uses an analytic expression for the eigenvalue sensitivities, but the aerodynamic matrix is computed analytically. All three methods are found to be in good agreement with each other. The sensitivities of the eigenvalues were used to predict the flutter speed, frequency, and reduced frequency. These approximations were found to be in good agreement with those obtained using a complete reanalysis.

The Modern Origin of Matrices and Their Applications

ERIC Educational Resources Information Center

Debnath, L.

2014-01-01

This paper deals with the modern development of matrices, linear transformations, quadratic forms and their applications to geometry and mechanics, eigenvalues, eigenvectors and characteristic equations with applications. Included are the representations of real and complex numbers, and quaternions by matrices, and isomorphism in order to show…

Introduction to the Application of the Dynalist Computer Program to the Analysis of Rail Systems Dynamics

DOT National Transportation Integrated Search

1974-08-01

DYNALIST, a computer program that extracts complex eigenvalues and eigenvectors for dynamic systems described in terms of matrix equations of motion, has been acquired and made operational at TSC. In this report, simple dynamic systems are used to de...

Linear quadratic regulators with eigenvalue placement in a horizontal strip

NASA Technical Reports Server (NTRS)

Shieh, Leang S.; Dib, Hani M.; Ganesan, Sekar

1987-01-01

A method for optimally shifting the imaginary parts of the open-loop poles of a multivariable control system to the desirable closed-loop locations is presented. The optimal solution with respect to a quadratic performance index is obtained by solving a linear matrix Liapunov equation.

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schunert, Sebastian; Wang, Yaqi; Gleicher, Frederick

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form ismore » based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.« less

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

DOE PAGES

Schunert, Sebastian; Wang, Yaqi; Gleicher, Frederick; ...

2017-02-21

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form ismore » based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.« less

Compressed modes for variational problems in mathematical physics and compactly supported multiresolution basis for the Laplace operator

NASA Astrophysics Data System (ADS)

Ozolins, Vidvuds; Lai, Rongjie; Caflisch, Russel; Osher, Stanley

2014-03-01

We will describe a general formalism for obtaining spatially localized (``sparse'') solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger's equation in quantum mechanics. Sparsity is achieved by adding an L1 regularization term to the variational principle, which is shown to yield solutions with compact support (``compressed modes''). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size. In addition, we introduce an L1 regularized variational framework for developing a spatially localized basis, compressed plane waves (CPWs), that spans the eigenspace of a differential operator, for instance, the Laplace operator. Our approach generalizes the concept of plane waves to an orthogonal real-space basis with multiresolution capabilities. Supported by NSF Award DMR-1106024 (VO), DOE Contract No. DE-FG02-05ER25710 (RC) and ONR Grant No. N00014-11-1-719 (SO).

Analytical development of disturbed matrix eigenvalue problem applied to mixed convection stability analysis in Darcy media

NASA Astrophysics Data System (ADS)

Hamed, Haikel Ben; Bennacer, Rachid

2008-08-01

This work consists in evaluating algebraically and numerically the influence of a disturbance on the spectral values of a diagonalizable matrix. Thus, two approaches will be possible; to use the theorem of disturbances of a matrix depending on a parameter, due to Lidskii and primarily based on the structure of Jordan of the no disturbed matrix. The second approach consists in factorizing the matrix system, and then carrying out a numerical calculation of the roots of the disturbances matrix characteristic polynomial. This problem can be a standard model in the equations of the continuous media mechanics. During this work, we chose to use the second approach and in order to illustrate the application, we choose the Rayleigh-Bénard problem in Darcy media, disturbed by a filtering through flow. The matrix form of the problem is calculated starting from a linear stability analysis by a finite elements method. We show that it is possible to break up the general phenomenon into other elementary ones described respectively by a disturbed matrix and a disturbance. A good agreement between the two methods was seen. To cite this article: H.B. Hamed, R. Bennacer, C. R. Mecanique 336 (2008).

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem

NASA Astrophysics Data System (ADS)

Chuluunbaatar, O.; Gusev, A. A.; Vinitsky, S. I.; Abrashkevich, A. G.

2009-08-01

A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675; O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685-693]. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials, a 3D-model of a hydrogen atom in a homogeneous magnetic field and a hydrogen atom on a three-dimensional sphere. Program summaryProgram title: ODPEVP Catalogue identifier: AEDV_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3001 No. of bytes in distributed program, including test data, etc.: 24 195 Distribution format: tar.gz Programming language: FORTRAN 77 Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP RAM: depends on the number and order of finite elements; the number of points; and the number of eigenfunctions required. Test run requires 4 MB Classification: 2.1, 2.4 External routines: GAULEG [3] Nature of problem: The three-dimensional boundary problem for the elliptic partial differential equation with an axial symmetry similar to the Schrödinger equation with the Coulomb and transverse oscillator potentials is reduced to the two-dimensional one. The latter finds wide applications in modeling of photoionization and recombination of oppositively charged particles (positrons, antiprotons) in the magnet-optical trap [4], optical absorption in quantum wells [5], and channeling of likely charged particles in thin doped films [6,7] or neutral atoms and molecules in artificial waveguides or surfaces [8,9]. In the adiabatic approach [10] known in mathematics as Kantorovich method [11] the solution of the two-dimensional elliptic partial differential equation is expanded over basis functions with respect to the fast variable (for example, angular variable) and depended on the slow variable (for example, radial coordinate ) as a parameter. An averaging of the problem by such a basis leads to a system of the second-order ordinary differential equations which contain potential matrix elements and the first-derivative coupling terms (see, e.g., [12,13,14]). The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program developed calculates potential matrix elements - integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter. These matrix elements can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [1,2]. Solution method: The parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions is solved by the finite element method using high-order accuracy approximations [15]. The generalized algebraic eigenvalue problem AF=EBF with respect to a pair of unknown ( E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [16]. First derivatives of the eigenfunctions with respect to the parameter which contained in potential matrix elements of the coupled system equations are obtained by solving the inhomogeneous algebraic equations. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials described in [1,17,18], a 3D-model of a hydrogen atom in a homogeneous magnetic field described in [14,19] and a hydrogen atom on a three-dimensional sphere [20]. Restrictions: The computer memory requirements depend on: the number and order of finite elements; the number of points; and the number of eigenfunctions required. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see sections below and listing for details). The user must also supply DOUBLE PRECISION functions POTCCL and POTCC1 for evaluating potential function U(ρ,z) of Eq. (1) and its first derivative with respect to parameter ρ. The user should supply DOUBLE PRECISION functions F1FUNC and F2FUNC that evaluate functions f(z) and f(z) of Eq. (1). The user must also supply subroutine BOUNCF for evaluating the parametric third type boundary conditions. Running time: The running time depends critically upon: the number and order of finite elements; the number of points on interval [z,z]; and the number of eigenfunctions required. The test run which accompanies this paper took 2 s with calculation of matrix potentials on the Intel Pentium IV 2.4 GHz. References:O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649-675 O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Comm. 179 (2008) 685-693. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986. O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, V.L. Derbov, L.A. Melnikov, V.V. Serov, Phys. Rev. A 77 (2008) 034702-1-4. E.M. Kazaryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 28 (2005) 423-430. Yu.N. Demkov, J.D. Meyer, Eur. Phys. J. B 42 (2004) 361-365. P.M. Krassovitskiy, N.Zh. Takibaev, Bull. Russian Acad. Sci. Phys. 70 (2006) 815-818. V.S. Melezhik, J.I. Kim, P. Schmelcher, Phys. Rev. A 76 (2007) 053611-1-15. F.M. Pen'kov, Phys. Rev. A 62 (2000) 044701-1-4. M. Born, X. Huang, Dynamical Theory of Crystal Lattices, The Clarendon Press, Oxford, England, 1954. L.V. Kantorovich, V.I. Krylov, Approximate Methods of Higher Analysis, Wiley, New York, 1964. U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127;A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640. C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320. O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov, S.I. Vinitsky, J. Phys. A 40 (2007) 11485-11524. A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40-64. K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982. O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243-269. Yu.A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330-361. O. Chuluunbaatar, A.A. Gusev, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich, M.S. Kaschiev, V.V. Serov, Comput. Phys. Comm. 178 (2008) 301-330. A.G. Abrashkevich, M.S. Kaschiev, S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328-348.

Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model

NASA Astrophysics Data System (ADS)

Berglund, Nils; Landon, Damien

2012-08-01

We study the stochastic FitzHugh-Nagumo equations, modelling the dynamics of neuronal action potentials in parameter regimes characterized by mixed-mode oscillations. The interspike time interval is related to the random number of small-amplitude oscillations separating consecutive spikes. We prove that this number has an asymptotically geometric distribution, whose parameter is related to the principal eigenvalue of a substochastic Markov chain. We provide rigorous bounds on this eigenvalue in the small-noise regime and derive an approximation of its dependence on the system's parameters for a large range of noise intensities. This yields a precise description of the probability distribution of observed mixed-mode patterns and interspike intervals.

Asymptotic behavior and interpretation of virtual states: The effects of confinement and of basis sets

NASA Astrophysics Data System (ADS)

Boffi, Nicholas M.; Jain, Manish; Natan, Amir

2016-02-01

A real-space high order finite difference method is used to analyze the effect of spherical domain size on the Hartree-Fock (and density functional theory) virtual eigenstates. We show the domain size dependence of both positive and negative virtual eigenvalues of the Hartree-Fock equations for small molecules. We demonstrate that positive states behave like a particle in spherical well and show how they approach zero. For the negative eigenstates, we show that large domains are needed to get the correct eigenvalues. We compare our results to those of Gaussian basis sets and draw some conclusions for real-space, basis-sets, and plane-waves calculations.

Collisionless kinetic theory of oblique tearing instabilities

DOE PAGES

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-15

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Deep learning and the electronic structure problem

NASA Astrophysics Data System (ADS)

Mills, Kyle; Spanner, Michael; Tamblyn, Isaac

In the past decade, the fields of artificial intelligence and computer vision have progressed remarkably. Supported by the enthusiasm of large tech companies, as well as significant hardware advances and the utilization of graphical processing units to accelerate computations, deep neural networks (DNN) are gaining momentum as a robust choice for many diverse machine learning applications. We have demonstrated the ability of a DNN to solve a quantum mechanical eigenvalue equation directly, without the need to compute a wavefunction, and without knowledge of the underlying physics. We have trained a convolutional neural network to predict the total energy of an electron in a confining, 2-dimensional electrostatic potential. We numerically solved the one-electron Schrödinger equation for millions of electrostatic potentials, and used this as training data for our neural network. Four classes of potentials were assessed: the canonical cases of the harmonic oscillator and infinite well, and two types of randomly generated potentials for which no analytic solution is known. We compare the performance of the neural network and consider how these results could lead to future advances in electronic structure theory.

Collisionless kinetic theory of oblique tearing instabilities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Optimized effective potential method and application to static RPA correlation

NASA Astrophysics Data System (ADS)

Fukazawa, Taro; Akai, Hisazumi

2015-03-01

The optimized effective potential (OEP) method is a promising technique for calculating the ground state properties of a system within the density functional theory. However, it is not widely used as its computational cost is rather high and, also, some ambiguity remains in the theoretical framework. In order to overcome these problems, we first introduced a method that accelerates the OEP scheme in a static RPA-level correlation functional. Second, the Krieger-Li-Iafrate (KLI) approximation is exploited to solve the OEP equation. Although seemingly too crude, this approximation did not reduce the accuracy of the description of the magnetic transition metals (Fe, Co, and Ni) examined here, the magnetic properties of which are rather sensitive to correlation effects. Finally, we reformulated the OEP method to render it applicable to the direct RPA correlation functional and other, more precise, functionals. Emphasis is placed on the following three points of the discussion: (i) level-crossing at the Fermi surface is taken into account; (ii) eigenvalue variations in a Kohn-Sham functional are correctly treated; and (iii) the resultant OEP equation is different from those reported to date.

Collisionless kinetic theory of oblique tearing instabilities

NASA Astrophysics Data System (ADS)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-01

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. We find that this stabilization is associated with the density-gradient-driven diamagnetic drift. The analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. A simple analytic estimate for the stability criterion is provided.

Normal modes of the shallow water system on the cubed sphere

NASA Astrophysics Data System (ADS)

Kang, H. G.; Cheong, H. B.; Lee, C. H.

2017-12-01

Spherical harmonics expressed as the Rossby-Haurwitz waves are the normal modes of non-divergent barotropic model. Among the normal modes in the numerical models, the most unstable mode will contaminate the numerical results, and therefore the investigation of normal mode for a given grid system and a discretiztaion method is important. The cubed-sphere grid which consists of six identical faces has been widely adopted in many atmospheric models. This grid system is non-orthogonal grid so that calculation of the normal mode is quiet challenge problem. In the present study, the normal modes of the shallow water system on the cubed sphere discretized by the spectral element method employing the Gauss-Lobatto Lagrange interpolating polynomials as orthogonal basis functions is investigated. The algebraic equations for the shallow water equation on the cubed sphere are derived, and the huge global matrix is constructed. The linear system representing the eigenvalue-eigenvector relations is solved by numerical libraries. The normal mode calculated for the several horizontal resolution and lamb parameters will be discussed and compared to the normal mode from the spherical harmonics spectral method.

Vibrational Modes of Oblate Clouds of Charge

NASA Astrophysics Data System (ADS)

Jenkins, Thomas; Spencer, Ross L.

2000-10-01

When a nonneutral plasma confined in a Penning trap is allowed time to expand, its shape at global thermal equilibrium is that of a thin oblate spheroid [D. L. Paulson et al., Phys. Plasmas 5, 345 (1998)]. Oscillations similar to those of a drumhead can be externally induced in such a plasma. Although a theory developed by Dubin predicts the frequencies of the various normal modes of oscillation [Phys. Rev. Lett. 66, 2076 (1991)], this theory assumes that the plasma has zero temperature and is confined by an ideal quadrupole electric field. Neither of these conditions is strictly true in experiments [C. S. Weimer et al., Phys. Rev. A 49, 3842 (1994)] where physical properties of the plasma are deduced from measurements of these frequencies, causing the measurements and ideal theory to differ by about 20%. We reformulate the problem of the normal oscillatory modes as a principal-value integral eigenvalue equation, including finite-temperature and non-ideal confinement effects. The equation is solved numerically to obtain the plasma's normal mode frequencies and shapes; reasonable agreement with experiment is obtained.

Eigensolver for a Sparse, Large Hermitian Matrix

NASA Technical Reports Server (NTRS)

Tisdale, E. Robert; Oyafuso, Fabiano; Klimeck, Gerhard; Brown, R. Chris

2003-01-01

A parallel-processing computer program finds a few eigenvalues in a sparse Hermitian matrix that contains as many as 100 million diagonal elements. This program finds the eigenvalues faster, using less memory, than do other, comparable eigensolver programs. This program implements a Lanczos algorithm in the American National Standards Institute/ International Organization for Standardization (ANSI/ISO) C computing language, using the Message Passing Interface (MPI) standard to complement an eigensolver in PARPACK. [PARPACK (Parallel Arnoldi Package) is an extension, to parallel-processing computer architectures, of ARPACK (Arnoldi Package), which is a collection of Fortran 77 subroutines that solve large-scale eigenvalue problems.] The eigensolver runs on Beowulf clusters of computers at the Jet Propulsion Laboratory (JPL).

Fast, purely growing collisionless reconnection as an eigenfunction problem related to but not involving linear whistler waves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bellan, Paul M.

If either finite electron inertia or finite resistivity is included in 2D magnetic reconnection, the two-fluid equations become a pair of second-order differential equations coupling the out-of-plane magnetic field and vector potential to each other to form a fourth-order system. The coupling at an X-point is such that out-of-plane even-parity electric and odd-parity magnetic fields feed off each other to produce instability if the scale length on which the equilibrium magnetic field changes is less than the ion skin depth. The instability growth rate is given by an eigenvalue of the fourth-order system determined by boundary and symmetry conditions. Themore » instability is a purely growing mode, not a wave, and has growth rate of the order of the whistler frequency. The spatial profile of both the out-of-plane electric and magnetic eigenfunctions consists of an inner concave region having extent of the order of the electron skin depth, an intermediate convex region having extent of the order of the equilibrium magnetic field scale length, and a concave outer exponentially decaying region. If finite electron inertia and resistivity are not included, the inner concave region does not exist and the coupled pair of equations reduces to a second-order differential equation having non-physical solutions at an X-point.« less

Coherent mode decomposition using mixed Wigner functions of Hermite-Gaussian beams.

PubMed

Tanaka, Takashi

2017-04-15

A new method of coherent mode decomposition (CMD) is proposed that is based on a Wigner-function representation of Hermite-Gaussian beams. In contrast to the well-known method using the cross spectral density (CSD), it directly determines the mode functions and their weights without solving the eigenvalue problem. This facilitates the CMD of partially coherent light whose Wigner functions (and thus CSDs) are not separable, in which case the conventional CMD requires solving an eigenvalue problem with a large matrix and thus is numerically formidable. An example is shown regarding the CMD of synchrotron radiation, one of the most important applications of the proposed method.

Numerical methods for the unsymmetric tridiagonal eigenvalue problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1996-12-31

This report summarizes the results of our project {open_quotes}Numerical Methods for the Unsymmetric Tridiagonal Eigenvalue Problem{close_quotes}. It was funded by both by a DOE grant (No. DE-FG02-92ER25122, 6/1/92-5/31/94, $100,000) and by an NSF Research Initiation Award (No. CCR-9109785, 7/1/91-6/30/93, $46,564.) The publications resulting from that project during the DOE funding period are listed below. Two other journal papers and two other conference papers were produced during the NSF funding period. Most of the listed conference papers are early or partial versions of the listed journal papers.

Capabilities of Fully Parallelized MHD Stability Code MARS

NASA Astrophysics Data System (ADS)

Svidzinski, Vladimir; Galkin, Sergei; Kim, Jin-Soo; Liu, Yueqiang

2016-10-01

Results of full parallelization of the plasma stability code MARS will be reported. MARS calculates eigenmodes in 2D axisymmetric toroidal equilibria in MHD-kinetic plasma models. Parallel version of MARS, named PMARS, has been recently developed at FAR-TECH. Parallelized MARS is an efficient tool for simulation of MHD instabilities with low, intermediate and high toroidal mode numbers within both fluid and kinetic plasma models, implemented in MARS. Parallelization of the code included parallelization of the construction of the matrix for the eigenvalue problem and parallelization of the inverse vector iterations algorithm, implemented in MARS for the solution of the formulated eigenvalue problem. Construction of the matrix is parallelized by distributing the load among processors assigned to different magnetic surfaces. Parallelization of the solution of the eigenvalue problem is made by repeating steps of the MARS algorithm using parallel libraries and procedures. Parallelized MARS is capable of calculating eigenmodes with significantly increased spatial resolution: up to 5,000 adapted radial grid points with up to 500 poloidal harmonics. Such resolution is sufficient for simulation of kink, tearing and peeling-ballooning instabilities with physically relevant parameters. Work is supported by the U.S. DOE SBIR program.

Fully Parallel MHD Stability Analysis Tool

NASA Astrophysics Data System (ADS)

Svidzinski, Vladimir; Galkin, Sergei; Kim, Jin-Soo; Liu, Yueqiang

2015-11-01

Progress on full parallelization of the plasma stability code MARS will be reported. MARS calculates eigenmodes in 2D axisymmetric toroidal equilibria in MHD-kinetic plasma models. It is a powerful tool for studying MHD and MHD-kinetic instabilities and it is widely used by fusion community. Parallel version of MARS is intended for simulations on local parallel clusters. It will be an efficient tool for simulation of MHD instabilities with low, intermediate and high toroidal mode numbers within both fluid and kinetic plasma models, already implemented in MARS. Parallelization of the code includes parallelization of the construction of the matrix for the eigenvalue problem and parallelization of the inverse iterations algorithm, implemented in MARS for the solution of the formulated eigenvalue problem. Construction of the matrix is parallelized by distributing the load among processors assigned to different magnetic surfaces. Parallelization of the solution of the eigenvalue problem is made by repeating steps of the present MARS algorithm using parallel libraries and procedures. Results of MARS parallelization and of the development of a new fix boundary equilibrium code adapted for MARS input will be reported. Work is supported by the U.S. DOE SBIR program.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brabec, Jiri; Lin, Lin; Shao, Meiyue

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires halfmore » of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.« less

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

DOE PAGES

Brabec, Jiri; Lin, Lin; Shao, Meiyue; ...

2015-10-06

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires halfmore » of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.« less

Programmable Calculator Use in Undergraduate Dynamics, Vibrations, and Elementary Structures Courses.

ERIC Educational Resources Information Center

Cutchins, M. A.

1982-01-01

Presents programmable calculator solutions to selected problems, including area moments of inertia and principal values, the 2-D principal stress problem, C.G. and pitch inertia computations, 3-D eigenvalue problems, 3 DOF vibrations, and a complex flutter determinant. (SK)

A B-spline Galerkin method for the Dirac equation

NASA Astrophysics Data System (ADS)

Froese Fischer, Charlotte; Zatsarinny, Oleg

2009-06-01

The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y=-λy, that can also be written as a pair of first-order equations y=λz, z=-λy. Expanding both y(r) and z(r) in the B basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the B basis and z(r) in the dB/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method ( B,B) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.

Quasinormal modes of Reissner-Nordstrom black holes

NASA Technical Reports Server (NTRS)

Leaver, Edward W.

1990-01-01

A matrix-eigenvalue algorithm is presented for accurately computing the quasi-normal frequencies and modes of charged static blackholes. The method is then refined through the introduction of a continued-fraction step. The approach should generalize to a variety of nonseparable wave equations, including the Kerr-Newman case of charged rotating blackholes.

Particle Acceleration and Radiative Losses at Relativistic Shocks

NASA Astrophysics Data System (ADS)

Dempsey, P.; Duffy, P.

A semi-analytic approach to the relativistic transport equation with isotropic diffusion and consistent radiative losses is presented. It is based on the eigenvalue method first introduced in Kirk & Schneider [5]and Heavens & Drury [3]. We demonstrate the pitch-angle dependence of the cut-off in relativistic shocks.

CFD simulation of simultaneous monotonic cooling and surface heat transfer coefficient

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mihálka, Peter, E-mail: usarmipe@savba.sk; Matiašovský, Peter, E-mail: usarmat@savba.sk

The monotonic heating regime method for determination of thermal diffusivity is based on the analysis of an unsteady-state (stabilised) thermal process characterised by an independence of the space-time temperature distribution on initial conditions. At the first kind of the monotonic regime a sample of simple geometry is heated / cooled at constant ambient temperature. The determination of thermal diffusivity requires the determination rate of a temperature change and simultaneous determination of the first eigenvalue. According to a characteristic equation the first eigenvalue is a function of the Biot number defined by a surface heat transfer coefficient and thermal conductivity ofmore » an analysed material. Knowing the surface heat transfer coefficient and the first eigenvalue the thermal conductivity can be determined. The surface heat transport coefficient during the monotonic regime can be determined by the continuous measurement of long-wave radiation heat flow and the photoelectric measurement of the air refractive index gradient in a boundary layer. CFD simulation of the cooling process was carried out to analyse local convective and radiative heat transfer coefficients more in detail. Influence of ambient air flow was analysed. The obtained eigenvalues and corresponding surface heat transfer coefficient values enable to determine thermal conductivity of the analysed specimen together with its thermal diffusivity during a monotonic heating regime.« less

On functional determinants of matrix differential operators with multiple zero modes

NASA Astrophysics Data System (ADS)

Falco, G. M.; Fedorenko, Andrei A.; Gruzberg, Ilya A.

2017-12-01

We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of r× r matrix second order differential operators O with 0 < n ≤slant 2r linearly independent zero modes. We separately discuss the cases of the homogeneous Dirichlet boundary conditions, when the number of zero modes cannot exceed r, and the case of twisted boundary conditions, including the periodic and anti-periodic ones, when the number of zero modes is bounded above by 2r. In all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the n zero modes and other r-n or 2r-n (depending on the boundary conditions) solutions of the homogeneous equation O h=0 , in the spirit of Gel’fand-Yaglom approach. In instanton calculations, the contribution of the zero modes is taken into account by introducing the so-called collective coordinates. We show that there is a remarkable cancellation of a factor (involving scalar products of zero modes) between the Jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues. This cancellation drastically simplifies instanton calculations when one uses our formulas.

Workshop report on large-scale matrix diagonalization methods in chemistry theory institute

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bischof, C.H.; Shepard, R.L.; Huss-Lederman, S.

The Large-Scale Matrix Diagonalization Methods in Chemistry theory institute brought together 41 computational chemists and numerical analysts. The goal was to understand the needs of the computational chemistry community in problems that utilize matrix diagonalization techniques. This was accomplished by reviewing the current state of the art and looking toward future directions in matrix diagonalization techniques. This institute occurred about 20 years after a related meeting of similar size. During those 20 years the Davidson method continued to dominate the problem of finding a few extremal eigenvalues for many computational chemistry problems. Work on non-diagonally dominant and non-Hermitian problems asmore » well as parallel computing has also brought new methods to bear. The changes and similarities in problems and methods over the past two decades offered an interesting viewpoint for the success in this area. One important area covered by the talks was overviews of the source and nature of the chemistry problems. The numerical analysts were uniformly grateful for the efforts to convey a better understanding of the problems and issues faced in computational chemistry. An important outcome was an understanding of the wide range of eigenproblems encountered in computational chemistry. The workshop covered problems involving self- consistent-field (SCF), configuration interaction (CI), intramolecular vibrational relaxation (IVR), and scattering problems. In atomic structure calculations using the Hartree-Fock method (SCF), the symmetric matrices can range from order hundreds to thousands. These matrices often include large clusters of eigenvalues which can be as much as 25% of the spectrum. However, if Cl methods are also used, the matrix size can be between 10{sup 4} and 10{sup 9} where only one or a few extremal eigenvalues and eigenvectors are needed. Working with very large matrices has lead to the development of« less

Analytical approach to peel stresses in bonded composite stiffened panels

NASA Technical Reports Server (NTRS)

Barkey, Derek A.; Madan, Ram C.; Sutton, Jason O.

1987-01-01

A closed-form solution was obtained for the stresses and displacements of two bonded beams. A system of two fourth-order and two second-order differential equations with the associated boundary equations was determined using a variational work approach. A FORTRAN computer program was devised to solve for the eigenvalues and eigenvectors of this system and to calculate the coefficients from the boundary conditions. The results were then compared with NASTRAN finite-element solutions and shown to agree closely.

The influence of thermal and conductive temperatures in a nanoscale resonator

NASA Astrophysics Data System (ADS)

Hobiny, Aatef; Abbas, Ibrahim A.

2018-06-01

In this work, the thermoelastic interaction in a nano-scale resonator based on two-temperature Green-Naghdi model is established. The nanoscale resonator ends were simply supported. In the Laplace's domain, the analytical solution of conductivity temperature and thermodynamic temperature, the displacement and the stress components are obtained. The eigenvalue approach resorted to for solutions. In the vector-matrix differential equations form, the essential equations were written. The numerical results for all variables are presented and are illustrated graphically.

Dynamics of the Smooth Positons of the Wadati-Konno-Ichikawa Equation

NASA Astrophysics Data System (ADS)

Wang, Gai-Hua; Zhang, Yong-Shuai; He, Jing-Song

2018-03-01

We discuss a modified Wadati-Konno-Ichikawa (mWKI) equation, which is equivalent to the WKI equation by a hodograph transformation. The explicit formula of degenerated solution of mWKI equation is provided by using degenerate Darboux transformation with respect to the eigenvalues, which yields two kinds of smooth solutions possessing the vanishing and nonvanishing boundary conditions respectively. In particular, a method for the decomposition of modulus square is operated to the positon solution, and the approximate orbits before and after collision of positon solutions are displayed explicitly. Supported by the National Natural Science Foundation of China under Grant No. 11671219, the K. C. Wong Magna Fund in Ningbo University

Stability of stagnation via an expanding accretion shock wave

NASA Astrophysics Data System (ADS)

Velikovich, A. L.; Murakami, M.; Taylor, B. D.; Giuliani, J. L.; Zalesak, S. T.; Iwamoto, Y.

2016-05-01

Stagnation of a cold plasma streaming to the center or axis of symmetry via an expanding accretion shock wave is ubiquitous in inertial confinement fusion (ICF) and high-energy-density plasma physics, the examples ranging from plasma flows in x-ray-generating Z pinches [Maron et al., Phys. Rev. Lett. 111, 035001 (2013)] to the experiments in support of the recently suggested concept of impact ignition in ICF [Azechi et al., Phys. Rev. Lett. 102, 235002 (2009); Murakami et al., Nucl. Fusion 54, 054007 (2014)]. Some experimental evidence indicates that stagnation via an expanding shock wave is stable, but its stability has never been studied theoretically. We present such analysis for the stagnation that does not involve a rarefaction wave behind the expanding shock front and is described by the classic ideal-gas Noh solution in spherical and cylindrical geometry. In either case, the stagnated flow has been demonstrated to be stable, initial perturbations exhibiting a power-law, oscillatory or monotonic, decay with time for all the eigenmodes. This conclusion has been supported by our simulations done both on a Cartesian grid and on a curvilinear grid in spherical coordinates. Dispersion equation determining the eigenvalues of the problem and explicit formulas for the eigenfunction profiles corresponding to these eigenvalues are presented, making it possible to use the theory for hydrocode verification in two and three dimensions.

Stability of stagnation via an expanding accretion shock wave

DOE Office of Scientific and Technical Information (OSTI.GOV)

Velikovich, A. L.; Giuliani, J. L.; Murakami, M.

Stagnation of a cold plasma streaming to the center or axis of symmetry via an expanding accretion shock wave is ubiquitous in inertial confinement fusion (ICF) and high-energy-density plasma physics, the examples ranging from plasma flows in x-ray-generating Z pinches [Maron et al., Phys. Rev. Lett. 111, 035001 (2013)] to the experiments in support of the recently suggested concept of impact ignition in ICF [Azechi et al., Phys. Rev. Lett. 102, 235002 (2009); Murakami et al., Nucl. Fusion 54, 054007 (2014)]. Some experimental evidence indicates that stagnation via an expanding shock wave is stable, but its stability has never beenmore » studied theoretically. We present such analysis for the stagnation that does not involve a rarefaction wave behind the expanding shock front and is described by the classic ideal-gas Noh solution in spherical and cylindrical geometry. In either case, the stagnated flow has been demonstrated to be stable, initial perturbations exhibiting a power-law, oscillatory or monotonic, decay with time for all the eigenmodes. This conclusion has been supported by our simulations done both on a Cartesian grid and on a curvilinear grid in spherical coordinates. Dispersion equation determining the eigenvalues of the problem and explicit formulas for the eigenfunction profiles corresponding to these eigenvalues are presented, making it possible to use the theory for hydrocode verification in two and three dimensions.« less

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

NASA Astrophysics Data System (ADS)

Gituliar, Oleksandr; Magerya, Vitaly

2017-10-01

We present Fuchsia - an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ∂x J(x , ɛ) = A(x , ɛ) J(x , ɛ) finds a basis transformation T(x , ɛ) , i.e., J(x , ɛ) = T(x , ɛ) J‧(x , ɛ) , such that the system turns into the epsilon form : ∂xJ‧(x , ɛ) = ɛ S(x) J‧(x , ɛ) , where S(x) is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator ɛ. That makes the construction of the transformation T(x , ɛ) crucial for obtaining solutions of the initial system. In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals. Program Files doi:http://dx.doi.org/10.17632/zj6zn9vfkh.1 Licensing provisions: MIT Programming language:Python 2.7 Nature of problem: Feynman master integrals may be calculated from solutions of a linear system of differential equations with rational coefficients. Such a system can be easily solved as an ɛ-series when its epsilon form is known. Hence, a tool which is able to find the epsilon form transformations can be used to evaluate Feynman master integrals. Solution method: The solution method is based on the Lee algorithm (Lee, 2015) which consists of three main steps: fuchsification, normalization, and factorization. During the fuchsification step a given system of differential equations is transformed into the Fuchsian form with the help of the Moser method (Moser, 1959). Next, during the normalization step the system is transformed to the form where eigenvalues of all residues are proportional to the dimensional regulator ɛ. Finally, the system is factorized to the epsilon form by finding an unknown transformation which satisfies a system of linear equations. Additional comments including Restrictions and Unusual features: Systems of single-variable differential equations are considered. A system needs to be reducible to Fuchsian form and eigenvalues of its residues must be of the form n + m ɛ, where n is integer. Performance depends upon the input matrix, its size, number of singular points and their degrees. It takes around an hour to reduce an example 74 × 74 matrix with 20 singular points on a PC with a 1.7 GHz Intel Core i5 CPU. An additional slowdown is to be expected for matrices with complex and/or irrational singular point locations, as these are particularly difficult for symbolic algebra software to handle.

Computation of eigenpairs of Ax = lambda Bx for vibrations of spinning deformable bodies

NASA Technical Reports Server (NTRS)

Utku, S.; Clemente, J. L. M.

1984-01-01

It is shown that, when linear theory is used, the general eigenvalue problem related with the free vibrations of spinning deformable bodies is of the type AX = lambda Bx, where A is Hermitian, and B is real positive definite. Since the order n of the matrices may be large, and A and B are banded or block banded, due to the economics of the numerical solution, one is interested in obtaining only those eigenvalues which fall within the frequency band of interest of the problem. The paper extends the well known method of bisections and iteration of R to the n power to n dimensional complex spaces, i.e., to C to the n power, so that it can be applied to the present problem.

Approximating the Helium Wavefunction in Positronium-Helium Scattering

NASA Technical Reports Server (NTRS)

DiRienzi, Joseph; Drachman, Richard J.

2003-01-01

In the Kohn variational treatment of the positronium- hydrogen scattering problem the scattering wave function is approximated by an expansion in some appropriate basis set, but the target and projectile wave functions are known exactly. In the positronium-helium case, however, a difficulty immediately arises in that the wave function of the helium target atom is not known exactly, and there are several ways to deal with the associated eigenvalue in formulating the variational scattering equations to be solved. In this work we will use the Kohn variational principle in the static exchange approximation to d e t e e the zero-energy scattering length for the Ps-He system, using a suite of approximate target functions. The results we obtain will be compared with each other and with corresponding values found by other approximation techniques.

Exact solution for four-order acousto-optic Bragg diffraction with arbitrary initial conditions.

PubMed

Pieper, Ron; Koslover, Deborah; Poon, Ting-Chung

2009-03-01

An exact solution to the four-order acousto-optic (AO) Bragg diffraction problem with arbitrary initial conditions compatible with exact Bragg angle incident light is developed. The solution, obtained by solving a 4th-order differential equation, is formalized into a transition matrix operator predicting diffracted light orders at the exit of the AO cell in terms of the same diffracted light orders at the entrance. It is shown that the transition matrix is unitary and that this unitary matrix condition is sufficient to guarantee energy conservation. A comparison of analytical solutions with numerical predictions validates the formalism. Although not directly related to the approach used to obtain the solution, it was discovered that all four generated eigenvalues from the four-order AO differential matrix operator are expressed simply in terms of Euclid's Divine Proportion.

Improvements to the fastex flutter analysis computer code

NASA Technical Reports Server (NTRS)

Taylor, Ronald F.

1987-01-01

Modifications to the FASTEX flutter analysis computer code (UDFASTEX) are described. The objectives were to increase the problem size capacity of FASTEX, reduce run times by modification of the modal interpolation procedure, and to add new user features. All modifications to the program are operable on the VAX 11/700 series computers under the VAX operating system. Interfaces were provided to aid in the inclusion of alternate aerodynamic and flutter eigenvalue calculations. Plots can be made of the flutter velocity, display and frequency data. A preliminary capability was also developed to plot contours of unsteady pressure amplitude and phase. The relevant equations of motion, modal interpolation procedures, and control system considerations are described and software developments are summarized. Additional information documenting input instructions, procedures, and details of the plate spline algorithm is found in the appendices.