Solution Of Eigenvalue Problem Research Articles

Effect of twist and rotation on vibration of functionally graded conical shells

This paper presents the effect of twist and rotational speeds on free vibration characteristics of functionally graded conical shells employing finite element method. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configuration of constituent materials on the frequencies. The equation of dynamic equilibrium is derived from Lagrange’s equation neglecting the Coriolis effect for moderate rotational speeds. The properties of the conical shell materials are presumed to vary continuously through their thickness with power-law distribution of the volume fractions of their constituents. The QR iteration algorithm is used for solution of standard eigenvalue problem. Computer codes developed are employed to obtain the numerical results concerning the combined effects of twist angle and rotational speed on the natural frequencies of functionally graded conical shells. The mode shapes for typical shells are also depicted. Numerical results obtained are the first known non-dimensional frequencies for the type of analyses carried out here.

Eigenvalue problems for nonlinear third‐order m‐point p‐Laplacian dynamic equations on time scales

This work deals with the existence and uniqueness of a nontrivial solution for the third‐order p‐Laplacian m‐point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when λ is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.

A multigrid method for eigenvalue problem

A multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem. In this scheme, solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem. Therefore, this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving. Some numerical experiments are presented to validate the efficiency of the method.

A novel local extrema based gravitational search algorithm and its application in face recognition using one training image per class

In this present paper a new methodology has been presented involving a stochastic optimization based approach to solve the face recognition problem with only one training image per class. Singular value decomposition (SVD) is used to decompose the single training image into two component images in order to compute the within class scatter matrix. The stochastic optimization approach is implemented employing gravitational search algorithm (GSA) which searches for an optimal transform matrix instead of using the traditional solution of general eigenvalue problem as is carried out in Fisher linear discriminant analysis (FLDA). The present paper also proposes two novel variants of GSA, namely the 2-D version of GSA, in order to cater for the 2-D image data, and the other one is a 2-D randomized local extrema based GSA (RLEGSA), which employs a stochastic local neighborhood based search instead of global search, as in basic GSA. Finally, a novel concept of performing an automated selection of projection vectors is incorporated in the 2-D RLEGSA to propose an improved variant, called the Modified RLEGSA (MRLEGSA). Experimental results, based on benchmark Yale A and ORL databases, show that the proposed methods outperform several existing schemes.

Acoustic wave propagation in inhomogeneous, layered waveguides based on modal expansions and hp-FEM

A new model is presented for harmonic wave propagation and scattering problems in non-uniform, stratified waveguides, governed by the Helmholtz equation. The method is based on a modal expansion, obtained by utilizing cross-section basis defined through the solution of vertical eigenvalue problems along the waveguide. The latter local basis is enhanced by including additional modes accounting for the effects of inhomogeneous boundaries and/or interfaces. The additional modes provide implicit summation of the slowly convergent part of the local-mode series, rendering the remaining part to be fast convergent, increasing the efficiency of the method, especially in long-range propagation applications. Using the enhanced representation, in conjunction with an energy-type variational principle, a coupled-mode system of equations is derived for the determination of the unknown modal-amplitude functions. In the case of multilayered environments, h- and p-FEM have been applied for the solution of both the local vertical eigenvalue problems and the resulting coupled mode system, exhibiting robustness and good rates of convergence. Numerical examples are presented in simple acoustic propagation problems, illustrating the role and significance of the additional mode(s) and the efficiency of the present model, that can be naturally extended to treat propagation and scattering problems in more complex 3D waveguides.

Light-Front Holography and the Light-Front Coupled-Cluster Method

We summarize the light-front coupled-cluster (LFCC) method for the solution of field-theoretic bound-state eigenvalue problems and indicate the connection with light-front holographic QCD. This includes a sample application of the LFCC method and leads to a relativistic quark model for mesons that adds longitudinal dynamics to the usual transverse light-front holographic Schrödinger equation.

An adaptive finite element method with asymptotic saturation for eigenvalue problems

This paper discusses adaptive finite element methods for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.

A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems (Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937-955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461-1483.) is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.

Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion

The computational efficiency of random field representations with the Karhunen–Loève (KL) expansion relies on the solution of a Fredholm integral eigenvalue problem. This contribution compares different methods that solve this problem. Focus is put on methods that apply to arbitrary shaped domains and arbitrary autocovariance functions. These include the Nyström method as well as collocation and Galerkin projection methods. Among the Galerkin methods, we investigate the finite element method (FEM) and propose the application of the finite cell method (FCM). This method is based on an extension to the FEM but avoids mesh generation on domains of complex geometric shape. The FCM was originally presented in Parvizian et al. (2007) [17] for the solution of elliptic boundary value problems. As an alternative to the L2-projection of the covariance function used in the Galerkin method, H1/2-projection and discrete projection are investigated. It is shown that the expansion optimal linear estimation (EOLE) method proposed in Li and Der Kiureghian (1993) [18] constitutes a special case of the Nyström method. It is found that the EOLE method is most efficient for the numerical solution of the KL expansion. The FEM and the FCM are more efficient than the EOLE method in evaluating a realization of the random field and, therefore, are suitable for problems in which the time spent in the evaluation of random field realizations has a major contribution to the overall runtime – e.g., in finite element reliability analysis.

Сдвиговые волны в упругом периодическом волноводе с альтернативнымн граничными условиями.

The propagation of shear waves in elastic waveguide of periodic structure consisting of three different materials with alternating along the guide walls boundary conditions is investigated. Using the transfer matrix approach the problem is reduced to the solution of a block transfer matrix eigenvalue problem. Bloth the dispersion and the band gap structure analysis have been carried out numerically. It is shown that for alternating boundary conditions along the waveguide walls, by modulating the ratio of the length of the unit cell to the width of the waveguide, the minimum widths of the stop bands can be moved to the middle of the Brillouin zone

Low-Rank Tensor Methods with Subspace Correction for Symmetric Eigenvalue Problems

We consider the solution of large-scale symmetric eigenvalue problems for which it is known that the eigenvectors admit a low-rank tensor approximation. Such problems arise, for example, from the discretization of high-dimensional elliptic PDE eigenvalue problems or in strongly correlated spin systems. Our methods are built on imposing low-rank (block) tensor train (TT) structure on the trace minimization characterization of the eigenvalues. The common approach of alternating optimization is combined with an enrichment of the TT cores by (preconditioned) gradients, as recently proposed by Dolgov and Savostyanov for linear systems. This can equivalently be viewed as a subspace correction technique. Several numerical experiments demonstrate the performance gains from using this technique.

NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems

A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear operator and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear operator has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independent of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small- and large-scale numerical examples are included.

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C 0 nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.

Time and frequency response of structures with frequency dependent, non-proportional linear damping

A method to compute the non-stationary time and frequency response of structures with a frequency-dependent non-proportional linear damping, called the resonance modes method, is presented in this paper. It consists of two main steps. The first step aims at spotting the structure resonance modes, which are the solutions of the matrix nonlinear eigenvalue problem obtained using the finite element method in the complex plane. This step requires a complex eigensolver and an iterative scheme, a perturbation technique or a combination of both. The second step uses the computed resonance modes and an analytical expression of the inverse Laplace transform to deduce the time or frequency response of structures to general excitations. The response of an aluminum plate damped with an elastomer treatment to a point-force excitation, computed with the classical modal approach, the direct solution and the presented method shows its precision and efficiency. An acoustic power computation finally validates the implementation of a fast variant, based on the perturbation technique, for vibroacoustic applications.

A higher order model for thin-walled structures with deformable cross-sections

A higher order model for the analysis of linear, prismatic thin-walled structures that considers the cross-section warping together with the cross-section in-plane flexural deformation is presented in this paper. The use of a one-dimentional model for the analysis of thin-walled structures, which have an inherent complex three-dimensional (3D) behaviour, can only be successful and competitive when compared with shell finite element models if it fulfills a twofold objective: (i) an enrichment of the model in order to as accurately as possible reproduce its 3D elasticity equations and (ii) the definition of a consistent criterion for uncoupling the beam equations, allowing to identify structural deformation modes.The displacement field is approximated through a linear combination of products between a set of linear independent functions defined over the cross-section and the associated weights only dependent on the beam axis; this approximation is not constrained by any ab initio kinematic assumptions. Towards an efficient application of the approximation procedure, the cross-section is discretized into thin-walled elements, being the displacement field approximated for each element independently of the displacement direction. The approximation is thus hp refined enhancing the “capture” of the 3D structural mechanics of thin-walled structures. The beam model governing equations are obtained through the integration over the cross-section of the corresponding elasticity equations weighted by the cross-section global approximation functions.A criterion for uncoupling the beam governing equations is established, allowing to (i) retrieve the classic equations of the thin-walled beam theory both for open and closed sections and (ii) derive a set of uncoupled deformation modes representing higher order effects. The criterion is based on the solution of the polynomial eigenvalue problem associated with the beam differential equations, allowing to quantify the Saint-Venant principle for thin-walled structures. In fact, the solution of the non linear eigenvalue problem yields a twelve fold null eigenvalue (representing polynomial solutions) that are verified to represent beam classic solutions and sets of pairs and quadruplets of non-null eigenvalues corresponding to higher order modes of deformation.

On three-dimensional global linear instability analysis of flows with standard aerodynamics codes

The development of a general Jacobian-free approach for the solution of large-scale global linear instability analysis eigenvalue problems by coupling a time-stepping algorithm with industry-standard second-order accurate aerodynamic codes is presented. The three-dimensional lid-driven cavity, a challenging flow in the context of required computational resources and physical complexity, has been chosen for validation. Results in excellent agreement with the literature have been obtained by using the proposed theoretical methodology coupled with the incompressible solver of the open-source toolbox OpenFOAM. The moderate computational resources required for the solution of the TriGlobal eigenvalue problem using this method opens up a new avenue for the performance of instability analysis of flows of engineering relevance.

Newton’s Method for the Solution of Inverse Spectral Problems

The inverse matrix eigenvalue problem is the problem of reconstruction of a matrix (determination of the unknown parameters of a certain matrix) for a given finite set of spectral data (points of the spectrum). The spectral data used in this case may contain complete or only partial information on the eigenvalues or eigenvectors. The aim of an inverse eigenvalue problem is to construct a matrix that preserves a certain specific structure and the prescribed spectral properties. There are two basic questions connected with any inverse eigenvalue problem: the theoretical question on the possibility of construction of its solutions and the practical question of its determination. The main investigations connected with solvability deal with establishing necessary or sufficient conditions under which the inverse eigenvalue problem is solvable. On the other hand, the main difficulty in calculations is connected with the development of a numerical procedure which would enable us to construct the matrix itself according to the a-priori specified spectral data. Both questions are important and complicated. Numerous works are devoted to the conditions of existence and uniqueness of solutions for different formulations of inverse matrix spectral problems and the methods used for their solution (see, e.g., [1–3, 7, 9–13, 15] and the literature therein). In the present work, we consider a numerical algorithm used for the solution of inverse eigenvalue problems under the assumption that this solution exists. Hence, we consider inverse algebraic eigenvalue problems.

A global result for discrete ϕ-Laplacian eigenvalue problems

Concerned is the existence, nonexistence and multiplicity of positive solutions for discrete ϕ-Laplacian eigenvalue problems. By using lower and upper solutions method and fixed point index theory, a global result with respect to parameter is established.

Interplay between discretization and algebraic computation in adaptive numerical solutionof elliptic PDE problems

AbstractThe Adaptive Finite Element Method (AFEM) for approximating solutions of PDE boundary value and eigenvalue problems is a numerical scheme that automatically and iteratively adapts the finite element space until a sufficiently accurate approximate solution is found. The adaptation process is based on a posteriori error estimators, and at each step of this process an algebraic problem (linear or nonlinear algebraic system or eigenvalue problem) has to be solved. In practical computations the solution of the algebraic problem cannot be obtained exactly. As a consequence, the algebraic error should be incorporated in the context of the AFEM and its a posteriori error estimators. The goal of this paper is to survey some existing approaches in the AFEM context that consider the interplay between the finite element discretization and the algebraic computation. We believe that a better understanding of this interplay is of great importance for the future development in the area of numerically solving large‐scale real‐world motivated problems. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximate soil-structure interaction analysis by a perturbation approach: The case of stiff soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a structure on an elastic soil is derived through the application of a perturbation analysis. For stiff soils, the full solution is obtained as the sum of the solution for a rigid-soil and small perturbing terms related to the inverse of the soil shear modulus. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented by other authors for the fundamental mode. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.