Approximate Eigensolutions Research Articles

Eigen-solutions and thermal properties of multi-parameter exponential potential

In this work, we determined an approximate eigen solutions of Modified multi-parameter exponential potential using supersymmetric quantum mechanics approach (SUSY) with improved Greene-Aldrich approximation to the centrifugal term. The energy equation and its corresponding normalised radial wave function were fully obtained. The proposed potential reduces to other useful potentials like Rosen-Morse, Hellmann, Yukawa and Coulomb potential as special cases. The thermodynamic properties like the vibrational mean energy (Uβ,V), Vibrational heat capacity (Cβ,V), vibrational entropy (Sβ,V) and vibrational free energy (Fβ,V) of the interacting potential were studied via partition function (Zβ,V) obtained from the resulting energy equation. This study was applied to three diatomic molecules: Chromium hydride (CrH), Titanium Hydride (TiH) and Thiocynate (ScN). To ascertain the high degree of our analytical mathematical accuracy, we compared the results of special cases with an existing results. These were found to be in excellent agreement with the existing results.

Improvement of the Envelope Theory for Systems with Different Particles

The envelope theory is a method to compute approximate eigensolutions of quantum $N$-body Hamiltonians with a quite general structure in $D$ dimensions. The advantages of the method are that it is easy to implement and that $N$ is treated as any other parameters of the Hamiltonian, allowing the computation for systems of all sizes. If solutions are reliable, they are generally not very accurate. In the case of systems with identical particles for $D \ge 2$, it is possible to improve the precision of the eigenvalues by combining the envelope theory with a generalisation to $N$-body of the dominantly orbital state method. It is shown that a similar improvement can be achieved in the case of systems composed of identical particles plus a different one. The quality of the new procedure is tested with different systems.

Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques

Numerical solution strategies for the Stokes eigenvalue problem based on the use of penalty formulations are investigated in this study. It is shown that the penalty method approach can successfully be adapted for the eigenproblem to rectify the associated problems such as the existence of zero diagonal entries in the resulting algebraic system. Two different schemes, namely, the standard penalisation with a small penalty parameter, and the iterative penalisation that enables relatively large parameters, are implemented. The employment of the latter leads to a so-called inhomogeneous generalised eigenvalue problem which requires a special attention. A feasible solution strategy is presented which is adapted from a procedure based on Newton's method proposed for the corresponding standard (inhomogeneous) eigenvalue problems. Concerning the spatial discretisation, among other possible options, the Chebyshev spectral collocation method based on expanding the unknown fields in tensor product of Chebyshev polynomials is employed. It is shown that the method constitutes a novel way of efficiently examining the approximate eigensolutions of the Stokes operator with the use of Chebyshev spectral collocation method directly, without a decoupling of velocity and pressure.

Efficient Approximate Eigensolution for Structures Equipped with a Passive Damping Device

AbstractThis article presents an approximate method for the complex eigensolution of nonclassically damped linear systems representing a structure equipped with a passive damping device. The propos...

Structural modal reanalysis for large, simultaneous and multiple type modifications

Eigenvalue problem is widely used to investigate the dynamic characteristics of large and complex structure. For finite element models, iterative solvers are needed to precisely calculate eigenvectors and eigenvalues. However, in cases such as large-scale reanalysis studies, or optimization design of huge structure, computational cost can quickly become too time consuming. This paper focus on the structural modal reanalysis for large and multiple modifications (including simultaneous boundary, topology and type modifications), which can greatly affect the eigen-modes of the modified structures. The proposed methods are based on the results from the modal analysis of the original structure, and the stiffness and mass matrix of the modified structures. A highly approximate eigensolution are generated by the proposed method, which is combined the newly added DOF (Degrees of Freedom) condensation, ICMO strategy (Independent and Coupling Mass Orthogonalization) with the Rayleigh–Ritz analysis. The numerical examples show that the proposed method is efficient with high precision even when large and multiple type modifications of the structural topology, boundary and type are made simultaneously.

Approximate eigensolutions of Dirac equation for the superposition Hellmann potential under spin and pseudospin symmetries

The Hellmann potential is simply a superposition of an attractive Coulomb poten- tial −a/r plus a Yukawa potential be −δr /r. The generalized parametric Nikiforov-Uvarov (NU) method is used to examine the approximate analytical energy eigenvalues and two-component wave function of the Dirac equation with the Hellmann potential for arbitrary spin-orbit quantum number κ in the presence of exact spin and pseudospin (p-spin) symmetries. As a particular case, we obtain the energy eigenvalues of the pure Coulomb potential in the non-relativistic limit.

Two Approximate Analytic Eigensolutions of the Hellmann Potential with any Arbitrary Angular Momentum

The parametric Nikiforov-Uvarov (pNU) and asymptotic iteration method (AIM) are applied to study the approximate analytic bound state eigensolutions (energy levels and wave functions) of the radial Schr¨odinger equation (SE) for the Hellmann potential which represents the superposition of the attractive Coulomb potential (-a/r) and the Yukawa potential bexp(-δ/r)/r of arbitrary strength b and screening parameter d in closed form. The analytical expressions to the energy eigenvalues Enl yield quite accurate results for a wide range of n; l in the limit of very weak screening but the results become gradually worse as the strength b and the screening coefficient d increase. The calculated bound state energies have been compared with available numerical data. Special cases of our solution like pure Coulomb and Yukawa potentials are also investigated.

Approximate Eigensolutions of the Deformed Woods—Saxon Potential via AIM

Using the Pekeris approximation, the Schrödinger equation is solved for the nuclear deformed Woods—Saxon potential within the framework of the asymptotic iteration method. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of hypergeometric function.

Approximate eigensolution of Laplacian matrices for locally modified graph products

Laplacian matrices and their spectrum are of great importance in algebraic graph theory. There exist efficient formulations for eigensolutions of the Laplacian matrices associated with a special class of graphs called product graphs. In this paper, the problem of determining a few approximate smallest eigenvalues and eigenvectors of large scale product graphs modified through the addition or deletion of some nodes and/or members, is investigated. The eigenproblem associated with a modified graph model is reduced using the set of master eigenvectors and linear approximated slave eigenvectors from the original model. Implicitly restarted Lanczos method is employed to obtain the required eigenpairs of the reduced problem. Examples of large scale models are included to demonstrate the efficiency of the proposed method compared to the direct application of the IRL method.

Approximate eigensolution of locally modified regular structures using a substructuring technique

A structure is called regular if its model can be considered as a product graph. A great computational advantage of regular structures, especially in the context of free vibration analysis, is their capacity for decomposition. In the present paper, the idea of decomposition is extended to the analysis of practical structural systems emerging as slight perturbations of the regular ones. A free interface dynamic substructuring method is employed to reformulate the eigenproblem as an assemblage of a base model and the modification part. The proposed method is very efficient in estimating a few lower eigenpairs of large-scale modified regular structures.

Energy spectra for quantum wires and two-dimensional electron gases in magnetic fields with Rashba and Dresselhaus spin-orbit interactions

We introduce an analytical approximation scheme to diagonalize parabolically confined two dimensional electron systems with both the Rashba and Dresselhaus spin-orbit interactions. The starting point of our perturbative expansion is a zeroth-order Hamiltonian for an electron confined in a quantum wire with an effective spin-orbit induced magnetic field along the wire, obtained by properly rotating the usual spin-orbit Hamiltonian. We find that the spin-orbit-related transverse coupling terms can be recast into two parts W and V, which couple crossing and non-crossing adjacent transverse modes, respectively. Interestingly, the zeroth-order Hamiltonian together with W can be solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics. We treat the V coupling by performing a Schrieffer-Wolff transformation. This allows us to obtain an effective Hamiltonian to third order in the coupling strength k_Rl of V, which can be straightforwardly diagonalized via an additional unitary transformation. We also apply our approach to other types of effective parabolic confinement, e.g., 2D electrons in a perpendicular magnetic field. To demonstrate the usefulness of our approximate eigensolutions, we obtain analytical expressions for the n^th Landau-level g_n-factors in the presence of both Rashba and Dresselhaus couplings. For small values of the bulk g-factors, we find that spin-orbit effects cancel out entirely for particular values of the spin-orbit couplings. By solving simple transcendental equations we also obtain the band minima of a Rashba-coupled quantum wire as a function of an external magnetic field. These can be used to describe Shubnikov-de Haas oscillations. This procedure makes it easier to extract the strength of the spin-orbit interaction in these systems via proper fitting of the data.

KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

Applying Eigenvalue Perturbation Theory to Solve Problems in Structural Dynamics

In this paper the eigenvalue perturbation theory is applied to solve various problems in structural dynamics. Detailed derivations are presented along with examples to demonstrate the utility of the approach and its accuracy. In undergraduate classes on mechanical vibration, the eigenvalue perturbation theory is rarely taught, but it can be extremely valuable. The approach can be used to analyze the effects of design changes on the modes of vibration of a dynamical system, to determine the approximate eigensolutions of a nearly uniform continuous system, and to obtain the approximate complex eigencharacteristics of a lightly damped system. This paper shows that the eigenvalue perturbation theory can be easily introduced at the undergraduate level, because to understand the derivations only a first course in linear algebra is required, well within the capability of an undergraduate engineering student.

KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

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 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 LDLT 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. 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,ρmax]; and (d) the number of eigensolutions required.

Approximate eigensolutions for arbitrarily damped nearly proportional systems

Two efficient methods for determining the approximate eigenvalues and eigenvectors for arbitrarily damped nearly proportional systems are developed. Both approaches are formulated by means of a first-order perturbation technique, whereby the real modes of vibration of the undamped system are used to derive approximate expressions for the complex eigenvalues and eigenvectors of a nearly proportionally damped system. Using either approach, the unperturbed configuration corresponds to a damped one whose damping matrix can be diagonalized by the same transformation that uncouples the undamped system, and the perturbation consists of the deviation of this diagonalizable damping matrix from the actual damping matrix. The proposed approaches are easy to code, implement and solve, and do not require forming state equations. Numerical examples are presented to validate the effectiveness of the current methods.

Domain Decomposition Spectral Approximations for an Eigenvalue Problem with a Piecewise Constant Coefficient

Consider a model eigenvalue problem with a piecewise constant coefficient. We split the domain at the discontinuity of the coefficient function and define the multidomain variational formulation for the eigenproblem. The discrete multidomain variational formulations are defined for Legendre--Galerkin and Legendre-collocation methods. The spectral rate of convergence of the approximate eigensolutions is proven for the Legendre--Galerkin method. The minmax principle is used for the convergence analysis.The Legendre-collocation, Chebyshev-collocation, Legendre-collocation penalty, and Chebyshev-collocation penalty methods are also defined by using the multidomain approach, and their numerical results applied to the eigenproblem are demonstrated. The spectral convergence for the eigenvalues and eigenfunctions is confirmed for all the multidomain spectral techniques presented here.

Approximate eigensolutions of axially moving beams with small flexural stiffness

Approximate eigensolutions of axially moving beams with small flexural stiffness

Eigensensitivity based optimal distribution of a viscoelastic damping layer for a flexible beam

In this paper, optimal distribution of a viscoelastic damping layer is sought for suppression of the transient vibration of a flexible beam. For the damping design, eigenvalues in the range of interest are taken as design criteria rather than the responses at a specific frequency. Two eigensensitivity based optimizing procedures are proposed, which are analogous to the pole placement technique and optimal control theory for dynamic system design. For the eigenanalysis of the structure with frequency-dependent material, Golla–Hughes–McTavish (GHM) model is used in expressing the viscoelastic material property and an approximate eigensolution is employed to avoid the intensity of iterative computation in the optimization process which is caused by additional degrees of freedom due to GHM modelling. Optimized partial coverage configurations are illustrated and compared to the full coverage configuration demonstrating the improved vibration characteristic of the optimally layered structure.

Two-step component-mode synthesis for the eigensolution of large systems

Component-mode synthesis is a dynamic substructuring technique for approximate eigensolutions of large degree-of-freedom systems divisible into two or more components. System synthesis using component modes results in approximate eigenparameters. The approximation is improved by using more component modes in synthesis. Typically, all component modes up to x times the frequency range of interest are required to ensure reasonable accuracy, where x varies from 2 to 10. A two-step component-mode synthesis approach is presented. The first step involves system synthesis using the minimum number of component modes required to obtain approximate cigenvalues up to a preselected frequency, and the second step uses additional component modes in a convergence scheme operating on the system eigenparameters calculated in the first step. This results in an infinite-series eigenvalue problem that is solved by a modified shooting method. A perturbation approach is adopted to derive the convergence scheme, in which separate iterations are performed for each eigenvalue and eigenvector in each step. Selection of initial values for the convergence scheme is also presented. In a few instances, an eigenvalue nearly reached the convergence tolerance, but then wandered away and converged to an adjacent value. The algorithm was modified to automatically detect this problem and abort iteration with a small penalty in accuracy. This two-step approach provides the analyst with some insight into the convergence and accuracy of the eigenparameters without resolving an increasingly larger system over and over with more component modes. Numerical examples are included to demonstrate the applicability of the method.

Accelerated reduction of subspace upper bound by multiple inverse iteration

The subspace iteration method is a popular and efficient approximate eigensolution technique for systems with a large number of degrees of freedom. This paper investigates the underlying physical concepts and mathematical background of the subspace iteration technique, and provides an explanation for the slow convergence rate of the higher required eigenpairs. A new acceleration approach, based on an estimate of each eigenvalue's ultimate rate of convergence, is developed and discussed. The method is implemented and tested on two sample numerical analyses, and compared to previous research by the authors. Research into combining this acceleration approach with other acceleration techniques is currently being performed, and shows much promise for producing a very efficient eigenproblem solving technique.