Solution Of Eigenvalue Problem Research Articles

Elastic stability of all edges clamped stepped and stiffened rectangular plate under uni-axial, bi-axial and shearing forces

The stability of clamped stepped and stiffened rectangular plate subjected to in-plane forces is examined. The plate is divided into 900 rectangular meshes and the partial derivatives are approximated using central difference formula. Altogether 841 equations of equilibrium and 248 equations representing boundary conditions are formed, finally leading to the solution of eigenvalue problem. The buckling coefficients are calculated for various types of stepped plates and the results are presented in tables for ready use by designers. The results are compared with the published results and they are in close agreement.

An algorithm for the complete solution of quadratic eigenvalue problems

We develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials. It incorporates scaling of the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable conditioning and backward stability properties, and a preprocessing step that reveals and deflates the zero and infinite eigenvalues contributed by singular leading and trailing matrix coefficients. The algorithm is backward-stable for quadratics that are not too heavily damped. Numerical experiments show that our MATLAB implementation of the algorithm, quadeig, outperforms the MATLAB function polyeig in terms of both stability and efficiency.

Adaptive-Multilevel BDDC and its parallel implementation

We combine the adaptive and multilevel approaches to the BDDC and formulate a method which allows an adaptive selection of constraints on each decomposition level. We also present a strategy for the solution of local eigenvalue problems in the adaptive algorithm using the LOBPCG method with a preconditioner based on standard components of the BDDC. The effectiveness of the method is illustrated on several engineering problems. It appears that the Adaptive-Multilevel BDDC algorithm is able to effectively detect troublesome parts on each decomposition level and improve convergence of the method. The developed open-source parallel implementation shows a good scalability as well as applicability to very large problems and core counts.

The unsteady flow of a weakly compressible fluid in a thin porous layer III: Three-dimensional computations

We describe a novel method for determining the pressure and velocity fields for a weakly compressible fluid flowing in a thin three-dimensional layer composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Our approach uses the method of matched asymptotic expansions to derive expressions for all significant process quantities, the computation of which requires only the solution of linear, elliptic, two-dimensional boundary-value and eigenvalue problems. In this article, we provide full implementation details and present numerical results demonstrating the efficiency and accuracy of our scheme.

The unsteady flow of a weakly compressible fluid in a thin porous layer II: Three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that ϵ ∝ h/l ≪ 1, we show that, to leading order in ϵ, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in ϵ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, ϵ, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

An a-posteriori error estimate for $$hp$$ -adaptive DG methods for elliptic eigenvalue problems on anisotropically refined meshes

We prove an a-posteriori error estimate for an \(hp\)-adaptive discontinuous Galerkin method for the numerical solution of elliptic eigenvalue problems with discontinuous coefficients on anisotropically refined rectangular elements. The estimate yields a global upper bound of the errors for both the eigenvalue and the eigenfunction and lower bound of the error for the eigenfunction only. The anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure. We present a series of numerical experiments to test the flexibility and robustness of this approach within a fully automated \(hp\)-adaptive refinement algorithm.

Effect of Multiple Delamination on Free Vibration Behaviour of Quasi-Isotropic Composite Conical Shells

In this paper, a finite element method is employed to investigate the free vibration characteristics of single and multiple delaminated graphite-epoxy quasi-isotropic composite conical shells. The generalized dynamic equilibrium equation is derived from Lagrange’s equation of motion neglecting Coriolis effect for moderate rotational speeds. The formulation is based on Mindlin’s theory considering eight-noded isoparametric plate bending element. The multipoint constraint algorithm is employed to ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. The QR iteration algorithm is utilized for solution of standard eigen value problem. Finite element codes are developed to obtain the natural frequencies of single and multiple delaminated quasi-isotropic composite conical shells. The mode shapes for a typical laminate configuration are also depicted. Numerical results obtained are the first known values which could serve as reference solutions for the future investigators.

ON DEVELOPMENT OF MULTISCALE FRACTURE MODELS

An approximate solution of nonlinear eigenvalue problem arising from the Mode I crack problem in a nonlinear power-law medium is obtained. The perturbation technique is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of constitutive law and eigenvalue corresponding to the linear undisturbed problem.

Wave propagation and attenuation in plates with periodic arrays of shunted piezo-patches

Periodic arrays of shunted piezoelectric patches are employed to control the wave propagation in a thin plate. The performance is characterized through the application of finite element method and Bloch theorem. This article proposes an effective approach to gain the dispersion properties of the periodically shunted plate in any directions from the solutions of transcendental eigenvalue problems. The results show that resistive shunts can tune the location and attenuation constants of the Bragg gap, while the internal resonances of resonant shunting system split the dispersion curves and form a locally resonant band gap. Moreover, the Bragg gap is directional, whose width varies enormously with directions. However, the locally resonant gap almost keeps the same in different directions, i.e., the gap is complete. Also, the influences of different shunting parameters to the band gaps are investigated in detail.

New sufficient conditions for robust stability analysis of interval matrices

This letter presents new sufficient conditions for robust Hurwitz stability of interval matrices. The proposed conditions are based on two approaches: (i) finding a common Lyapunov matrix for the interval family and (ii) converting the robust stability problem into a robust non-singularity problem using Kronecker operations. The main contribution of the letter is to derive accurate and computationally simple optimal estimates of the robustness margin and spectral bound of general interval matrices. The evaluation of the condition relies on the solutions of linear matrix inequalities (LMIs) and eigenvalue problems, both of which are solved very efficiently. The improvements gained by using the proposed conditions are demonstrated through application to previous examples in the literature.

Laplacians on a family of quadratic Julia sets II

This paper continues the work started in [4] to construct $P$-invariant Laplacians on the Julia sets of $P(z) = z^2 + c$ for $c$ in the interior of the Mandelbrot set, and to study the spectra of these Laplacians numerically. We are able to deal with a larger class of Julia sets and give a systematic method that reduces the construction of a $P$-invariant energy to the solution of nonlinear finite dimensional eigenvalue problem. We give the complete details for three examples, a dendrite, the airplane, and the Basilica-in-Rabbit. We also study the spectra of Laplacians on covering spaces and infinite blowups of the Julia sets. In particular, for a generic infinite blowups there is pure point spectrum, while for periodic covering spaces the spectrum is a mixture of discrete and continuous parts.

A time-domain Discontinuous Galerkin method for mechanical resonator quality factor computations

Numerical simulations are becoming increasingly important in the design of micromechanical resonators, in particular for the prediction of complex frequency response in high quality devices where damping is controlled by anchor losses. Frequency based approaches have been shown to predict these accurately, however, they require the solution of eigenvalue problems or the inversion of Helmholtz-type operators which are known to be very difficult for large-scale iterative solvers. We propose using a time-domain approach instead, where a broadband input signal is propagated through the system with a local explicit time-stepper. This is achieved using a new high-order Discontinuous Galerkin (DG) discretization for the linear elasticity equations, in particular a second-order formulation with Compact DG fluxes and a Runge–Kutta time integrator, where the block-diagonal mass matrices allow for efficient, stable, and accurate time stepping. Our solver scales well on distributed parallel computers, even in three spatial dimension and for large problem sizes. The resulting output signal is analyzed using a well-known filter diagonalization method, which is capable of finding accurate frequencies and quality factors for as little as a hundred periods of data. We validate the properties of our scheme on model problems, and demonstrate the feasibility of our proposed analysis process on two high quality factor disk resonators, using an axisymmetric formulation as well as full three dimensional simulations which is shown to scale well.

Existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations

In this article, by using the fixed point theorem, existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations D 0 + α u ( t ) + λ a ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 is considered, where 1 < α < 2 is a real number, D 0 + α is the standard Riemann-Liouville derivate, λ is a positive parameter and a(t) ∈ C([0, 1], [0, ∞)), f(t, u) ∈ C([0, 1] × [0, ∞), [0, ∞)).MSC(2010): 34B18.

Natural frequencies of delaminated composite rotating conical shells—A finite element approach

This paper presents the effect of rotational speeds on free vibration characteristics of delaminated twisted graphite–epoxy cross-ply composite conical shells employing finite element method. Theoretical formulation is based on Mindlin's theory considering an eight noded isoparametric plate bending element. A generalized dynamic equilibrium equation is derived from Lagrange's equation of motion neglecting the Coriolis effect for moderate rotational speeds. The multi-point constraint algorithm is utilized to ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. The QR iteration algorithm is used for solution of standard eigenvalue problem. Finite element codes are developed to obtain the numerical results concerning the combined effects of twist angle and rotational speed on the natural frequencies of cross-ply composite shallow conical shells. The mode shapes for a typical laminate configuration are also depicted. Numerical results obtained for cross-ply laminates with delamination are the first known non-dimensional frequencies for the type of analyses carried out here.

Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the existence and bound of solutions for some nonlinear equation. Moreover, for some nonlinear problems, we show that if the first p-eigenvalue of a domain is big, the supremum of a solution related to this domain is close to zero. For that we obtain L ∞ estimates for solutions of nonlinear and eigenvalue problems in terms of other L p norms.

A relaxation method for large eigenvalue problems, with an application to flow stability analysis

Linear stability analysis of fluid flows usually involves the numerical solution of large eigenvalue problems. We present a spectral transformation allowing the computation of the least stable eigenmodes in a prescribed frequency range, based on the filtering of the linearized equations of motion. This “shift-relax” method has the advantage of low memory requirements and is therefore suitable for large two- or three-dimensional problems. For demonstration purposes, this new method is applied to compute eigenmodes of a compressible jet.

Positive Solution of Singular Eigenvalue Problems for Third-Order Differential Equations

Positive Solution of Singular Eigenvalue Problems for Third-Order Differential Equations

One Approach to Construction of Bilateral Approximations Methods for Solution of Nonlinear Eigenvalue Problems

In this paper a new approach to construction of iterative methods of bilateral approximations of eigenvalue is proposed and investigated. The conditions on initial approximation, which ensure the convergence of iterative processes, are obtained.

Parallel Rayleigh Quotient Optimization with FSAI‐Based Preconditioning

The present paper describes a parallel preconditioned algorithm for the solution of partial eigenvalue problems for large sparse symmetric matrices, on parallel computers. Namely, we consider the Deflation‐Accelerated Conjugate Gradient (DACG) algorithm accelerated by factorized‐sparse‐approximate‐inverse‐ (FSAI‐) type preconditioners. We present an enhanced parallel implementation of the FSAI preconditioner and make use of the recently developed Block FSAI‐IC preconditioner, which combines the FSAI and the Block Jacobi‐IC preconditioners. Results onto matrices of large size arising from finite element discretization of geomechanical models reveal that DACG accelerated by these type of preconditioners is competitive with respect to the available public parallel hypre package, especially in the computation of a few of the leftmost eigenpairs. The parallel DACG code accelerated by FSAI is written in MPI‐Fortran 90 language and exhibits good scalability up to one thousand processors.

Positive Solutions of Eigenvalue Problems for a Class of Fractional Differential Equations with Derivatives

By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional differential equations is discussed. Some sufficient conditions for the existence of positive solutions are established.