Eigenvalue Solver Research Articles

Free vibration of doubly-curved generally laminated composite panels with viscoelastic matrix

As a first endeavor, this study scrutinizes the effects of general lamination schemes on the natural frequencies and loss factors of simply supported doubly-curved laminated composite panels. Transversely isotropic fibers obey linear elastic rule while isotropic polymeric matrix follows both standard solid and Kelvin-Voigt viscoelastic models. Frequency-dependent properties are obtained by multi-cell micromechanical approach and Alfrey correspondence principle. The governing equations of motions with frequency-dependent shear-stretching, bending-stretching and bending-twisting couplings are derived using Hamilton principle and third-order shear deformation theory. The coupled PDEs of motions are figured out via Galerkin method and eigenvalue solver. To verify, vibrational characteristics are compared with available results of elastic laminated panels and viscoelastic sandwich plates. Then, the effects of constitutive relations, fiber volume fraction, different fibers, geometrical parameters, symmetric, anti-symmetric and asymmetric laminations schemes are investigated on vibrational characteristics of spherical, cylindrical and hyperboloid panels. Also, crossing phenomenon and curve veering of frequencies and loss factors are observed.

Convergence study on ultrasonic guided wave propagation modes in an axisymmetric cylindrical waveguide

Guided wave propagation and its dispersion phenomenon of infinite solid elastic rods are encountered in several applications including, mechanical and civil engineering fields. In this paper, the elastic stress wave propagation in the axisymmetric circular cross-section of a high strength steel wire with cylindrical waveguide is investigated using a semi-analytical finite element (SAFE) method. The error analyses are carried out on fundamental modes, namely, flexural longitudinal and torsional modes. The theoretical framework for finite element (FE) discretization is established for the cylindrical waveguide. A three-node triangular linear element is used for solving SAFE dispersion solutions such as the wavenumber-frequency curve, phase velocity, and group velocity curves. The convergence and accuracy of the method are analyzed by comparing it with the calculation results of the transcendental Pochhammer frequency equation, and the meshing criterion is proposed. The use of higher-order (quadratic) elements are proposed for lower computational burden and effective method for solving eigenvalue problems. The effect of using 6-node triangular quadratic elements in the SAFE method for improving frequency accuracy is discussed in a detailed manner by proposing the meshing criterion. The calculation accuracy of a quadratic element semi-analytical discretization exceeds that of a linear element discretization with four times its radial circumference. The statistical histograms are demonstrated to prove the results of the proposed semi-analytical discretization method which can be used for solving cylindrical waveguides.

Spectral aspects of Nitsche’s method on nonconforming meshes

Nitsche’s method is an effective framework for the solution of problems involving embedded domains. Weak enforcement of Dirichlet boundary and interface conditions engenders additional active degrees of freedom compared to the corresponding kinematically admissible formulation, and hence additional solutions in eigenvalue problems. The original and added eigenpairs are designated proper and complementary, respectively. The number of complementary pairs equals the number of degrees of freedom that would be constrained in the kinematically admissible formulation. We investigate the number of complementary pairs that arise in representative nonconforming configurations of bilinear quadrilaterals. Algebraic elimination of the added degrees of freedom from the Nitsche formulation yields a formulation with several advantageous features. Practical procedures for solving eigenvalue problems based on reduced methods are proposed.

A generalized conjugate gradient method for eigenvalue problems

A generalized conjugate gradient method is proposed to solve eigenvalue problems. This method is designed bycombining the dumping block inverse power scheme and the subspace projection method. Furthermore, based on theproperties of the proposed method, a series of optimization techniques are developed toimprove the stability, computing efficiency and scalability. We also introducea computing package GCGE (generalized conjugate gradient eigensolver)which is developed based on the proposed method here.Some numerical examples are provided to validate the stability, computing efficiency and scalabilityof the method in this paper. The corresponding computing package can be downloaded from the website <uri xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=http://https://github.com/pase2017/GCGE-1.0>https://github.com/pase2017/GCGE-1.0</uri>

A query‐based quantum eigensolver

Solving eigenvalue problems is crucially important for both classical and quantum applications. Many well-known numerical eigensolvers have been developed, including the QR and the power methods for classical computers, as well as the quantum phase estimation(QPE) method and the variational quantum eigensolver for quantum computers. In this work, we present an alternative type of quantum method that uses fixed-point quantum search to solve Type II eigenvalue problems. It serves as an important complement to the QPE method, which is a Type I eigensolver. We find that the effectiveness of our method depends crucially on the appropriate choice of the initial state to guarantee a sufficiently large overlap with the unknown target eigenstate. We also show that the quantum oracle of our query-based method can be efficiently constructed for efficiently-simulated Hamiltonians, which is crucial for analyzing the total gate complexity. In addition, compared with the QPE method, our query-based method achieves a quadratic speedup in solving Type II problems.

Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman-Kac formula in terms of forward-backward stochastic differential equations. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neural-network ansatz. The criterion of fixed point provides a natural loss function to search for parameters via optimization. Our approach is able to provide accurate eigenvalue and eigenfunction approximations in several numerical examples, including Fokker-Planck operator and the linear and nonlinear Schrödinger operators in high dimensions.

Investigation of damping effects on low-frequency steady-state acoustical behaviour of coupled spaces.

In the low-frequency range, the acoustical behaviour of enclosed spaces is strongly influenced by excited acoustic modes resulting in a spatial irregularity of a steady-state sound field. In the paper, this problem has been examined theoretically and numerically for a system of coupled spaces with complex-valued conditions on boundary surfaces. Using a modal expansion method, an analytic formula for Green’s function was derived allowing to predict the interior sound field for a pure-tone excitation. To quantify the spatial irregularity of steady-state sound field, the parameter referred to as the mean spatial deviation was introduced. A numerical simulation was carried out for the system consisting of two coupled rectangular subspaces. Eigenfunctions and eigenfrequencies for this system were determined using the high-accuracy eigenvalue solver. As was evidenced by computational data, for small sound damping on absorptive walls the mean spatial deviation peaks at frequencies corresponding to eigenfrequencies of strongly localized modes. However, if the sound damping is much higher, the main cause of spatial irregularity of the interior sound field is the appearance of sharp valleys in a spatial distribution of a sound pressure level.

A unified algorithm for fully-coupled aeroelastic stability analysis of conical shells in yawed supersonic flow to identify the effect of boundary conditions

Supersonic flutter analysis of truncated conical shells with arbitrary classical boundary conditions is investigated in the present study. Structural equations of motion are obtained in their integral representation using the Flügge's first-order shear deformable assumptions along with the first-order piston theory for the aerodynamic model. Effects of curvature correction and airflow's yaw angle are also included in the aerodynamic model. Using a Rayleigh-Ritz based solution algorithm, the aeroelastic system of equations is solved by a standard eigenvalue solver with quadruple precision. After verification of the presented algorithm and solution procedure, a comprehensive parametric study is performed and effects of various parameters, namely the boundary conditions, semi-vertex angle, length and thickness ratios and yaw angle of the airflow on flutter boundaries of truncated conical shells are examined.

Wave propagation analysis of a rectangular sandwich composite plate with tunable magneto-rheological fluid core

This study presents an investigation of the wave propagation of a rectangular sandwich composite plate with tunable magneto-rheological fluid core. The constituent parts of this rectangular sandwich composite plate are the base layer, magneto-rheological fluid core, and limiter layer. Magneto-rheological fluid core is embedded within the base and limiter layers. Also, the upper and lower layers are made of elastic materials. For obtaining the governing equations of motion, Hamilton’s principle and classical plate theory are used. After that, applying an analytical solution, the wave frequency and phase velocity of the propagated waves can be gained by solving eigenvalue problem. By investigating the effect of the magnetic field, the results emphasize that the magnetic field intensity is the most important factor for changing the value of the wave frequency and phase velocity. Besides, results show by enhancing the core-to-top layer thickness ratio, the wave frequencies reduce because magneto-rheological fluid core is softer in comparison to elastic layers. Therefore, magneto-rheological fluid layer operates similar to a damper. At the end, the influence of the thickness of magneto-rheological fluid core on the phase velocity is also discussed.

Kinetic damping of radially localized kinetic toroidal Alfvén eigenmodes in tokamak plasmas

A global eigenvalue solver code is used to analyze the kinetic damping of radially localized kinetic toroidal Alfvén eigenmodes (KTAEs). By including the finite-Larmor-radius effects of ions, KTAEs are found in the Alfvén continuum well above the upper boundary of the TAE gap. The numerical calculations reveal that the real frequency and the kinetic damping of the KTAEs increase with increasing ion Larmor radius, with the ion-Larmor-radius effect intensifying for KTAEs with higher radial mode number. With increasing plasma β, the real frequency of the KTAEs increases whereas the damping rate remains almost constant. At an intermediate plasma β, the real frequencies, damping rates, and mode structures of each pair of KTAEs with radial mode numbers p and p + 1 merge into each other. The damping of KTAEs with lower radial mode number remains unchanged with the tokamak aspect ratio, whereas it increases for KTAEs with higher radial mode number. The KTAEs have lower kinetic damping and a broader mode structure near the core region of a tokamak having a flat core q equilibrium.

Wave propagation analysis of electro-rheological fluid-filled sandwich composite beam

In this article, wave propagation of the sandwich composite beam with tunable electro-rheological (ER) fluid core is investigated. The sandwich composite beam is made of three layers consisting of the basic layer, ER fluid core, and the limiter layer. ER fluid core embedded withindoors the basic and limiter layers. The upper and lower layers are constructed of the elastic materials. Hamilton’s principle is utilized for deriving the governing equations of motion. Using an analytical solution, the wave frequency and the phase velocity can be gathered by solving eigenvalue problem. Moreover, the effect of different parameters such as electric field, core-to-top layer thickness ratio, and thickness of ER core is investigated on the wave dispersion characteristics.

Multiscale modal analysis of fully-loaded spent nuclear fuel canisters

In this paper, an efficient numerical procedure is proposed, which is adapted to the modal analysis of a fully-loaded spent nuclear fuel canister exhibiting distinct structural levels associated with a hierarchy of components. On the one hand, the fully-loaded spent nuclear fuel canister is constituted of a repetition of identical components, resulting in a pseudo-periodicity; and the components of a given level are separated from each other and only connected to their upper level through localized attachments. This gives rise to an advantageous structural connectivity that can be exploited for efficiency. On the other hand, the necessary fine mesh resolution of the small levels leads to a high-dimensional computational model and, in addition, the independent resonant vibrations of each of the components produce a very large number of vibration eigenmodes. The aforementioned opportunities and difficulties are respectively leveraged and tackled by an adapted method that combines domain decomposition, shift–invert Lanczos eigenvalue solver, and Craig–Bampton substructuring technique. A parallel eigensolution via spectrum slicing is facilitated by an efficient block factorization by Schur complement due to the sparsity of the Craig–Bampton matrices. A computational gain of four orders of magnitude is obtained, at the expense of negligible errors that are exactly characterized. The proposed methodology enables a high-fidelity vibration analysis of the sealed fully-loaded spent nuclear fuel canister, useful for non-intrusive inverse identification of the structural integrity of the internal structural levels holding the nuclear material.

A fundamental understanding of the unstable eigenmodes of double tearing instabilities in a shear slab geometry

Two types of unstable eigenmodes of resistive tearing instabilities, namely, symmetric and anti-symmetric modes, in a double current sheet configuration are analyzed by means of both an eigenvalue solver and initial value simulation. It has been clearly identified that these two types of eigenmodes are different from the two independent single tearing modes even though the symmetric eigenmode in a double current sheet configuration shares the same properties of the single tearing mode near each current sheet. In the case with finite separation Δx between two current sheets, an arbitrary phase disturbance between two current sheets can lead to “phase instability,” i.e., the transition from the symmetric mode to the anti-symmetric mode. For a large Δx limit, both anti-symmetric and symmetric modes share the same properties of the single tearing mode. Thus, the superposition of two independent single tearing modes with arbitrary phases and arbitrary amplitudes at two current sheets can become the linear combination of symmetric and anti-symmetric eigenmodes. The same growth rate/eigenvalue of symmetric and anti-symmetric eigenmodes infers that an arbitrary superposition of two independent single tearing modes is still the eigenmode of the double current sheet configuration.

High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem

We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the Bethe-Salpeter equation involves the solution of a large, dense, skew-symmetric eigenvalue problem. The computed eigenpairs can be used to compute the optical absorption spectrum of molecules and crystalline systems. One state-of-the art high-performance solver package for symmetric matrices is the ELPA (Eigenvalue SoLvers for Petascale Applications) library. We exploit a link between tridiagonal skew-symmetric and symmetric matrices in order to extend the methods available in ELPA to skew-symmetric matrices. This way, the presented solution method can benefit from the optimizations available in ELPA that make it a well-established, efficient and scalable library. The solution strategy is to reduce a matrix to tridiagonal form, solve the tridiagonal eigenvalue problem and perform a back-transformation for eigenvectors of interest. ELPA employs a one-step or a two-step approach for the tridiagonalization of symmetric matrices. We adapt these to suit the skew-symmetric case. The two-step approach is generally faster as memory locality is exploited better. If all eigenvectors are required, the performance improvement is counteracted by the additional back transformation step. We exploit the symmetry in the spectrum of skew-symmetric matrices, such that only half of the eigenpairs need to be computed, making the two-step approach the favorable method. We compare performance and scalability of our method to the only available high-performance approach for skew-symmetric matrices, an indirect route involving complex arithmetic. In total, we achieve a performance that is up to 3.67 times higher than the reference method using Intel’s ScaLAPACK implementation. Our method is freely available in the current release of the ELPA library.

Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials

This paper analyzes spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps among the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local operator preconditioning by domain decomposition.

Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis

A new coupling interface degrees of freedom (DOFs) reduction technique for the mixed-interface component mode synthesis (MCMS) method is proposed, which referred to as the MCMS-rid method. This approach employs a set of shape functions via the linear interpolation (LI) in finite element method (FEM) to realize interface nodal coordinate transformations for each substructure, and then only a small number of interpolation basic nodes (IBNs) will be involved in mode synthesis and the following dynamic analysis. Unlike the majority of available CMS methods that retain a full dimension of the coupling interface DOFs, the MCMS-rid method allows to reduce the coupling interface DOFs significantly and enhance the computational efficiency. Three numerical models, including a rectangular beam with two ends fixed, a non-rectangular beam with the button fixed and a simplified dam-foundation system with different material properties, are presented to demonstrate the computational accuracy and efficiency of the proposed method. The results indicate that favourable accuracy with a least number of retained DOFs involved in mode synthesis can be obtained for solving eigenvalue problems when compared with other MCMS methods. The optimal number and distribution of the IBNs are discussed on structural dynamic analysis as well. It is shown that the more the IBNs are involved in mode synthesis, the better the precision that will be received. Furthermore, when the sub-regions are nearly square, the precision is best.

Variational quantum algorithms for dimensionality reduction and classification

In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil $(\mathcal{G},\mathcal{S})$. As a proof-of-principle, we implement our algorithm to solve $2^5\times2^5$ generalized eigenvalue problems. Finally, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.

Solution of Thermoacoustic Eigenvalue Problems With a Noniterative Method

Abstract Gas turbine combustors are prone to undesirable combustion dynamics in the form of thermoacoustic oscillations. Analysis of the stability of thermoacoustic systems in the frequency domain leads to nonlinear eigenvalue problems (NLEVP); here, “nonlinear” refers to the fact that the eigenvalue, the complex oscillation frequency, appears in a nonlinear fashion. In this paper, we employ a noniterative strategy based on contour integration in the complex eigenvalue plane, which returns all eigenvalues inside the contour. An introduction to the technique is given, and is complemented with guidelines for the specific application to thermoacoustic problems. Two prototypical nonlinear eigenvalue problems are considered: a network model of the classical Rijke tube with an analytic flame response model and a finite element discretization of an annular model combustor with an experimental flame transfer function (FTF). Computation of all eigenvalues in a domain of interest is vital to assess stability of these systems. We demonstrate that this is generally challenging for iterative strategies. An eigenvalue solver based on contour integration, in contrast, provides a reliable, noniterative method to achieve this goal.

Wave dispersion characteristics of a rectangular sandwich composite plate with tunable magneto-rheological fluid core rested on a visco-Pasternak foundation

In this paper, a sandwich composite plate with tunable magneto-rheological (MR) fluid core is utilized for analyzing the wave propagation. Sandwich composite plate is made of three layers consisting of the base layer, MR layer and the limiter layer. MR layer is embedded within the base and the limiter layers. Moreover, the upper and the lower layers are made of the elastic materials. This structure is also located on a viscoelastic substrate. At first, for achieving the governing equations of motion, Hamilton’s principle and classical plate theory (CLPT) are utilized. Then, exerting an analytical solution, the wave frequency and phase velocity of the propagated waves can be gained by solving eigenvalue problem. By investigating the effect of the magnetic field, the results emphasize that the magnetic field intensity is most important factor for changing the value of the wave dispersion characteristic. Besides, results indicate by enhancing the core-to-top layer thickness ratio, the wave frequency decreases because MR fluid core is softer in comparison to the elastic layers, Hence, MR fluid core operates similar damper. At the end, the effect of the other parameters such as thickness of MR layer, Winkler, Pasternak and damping coefficients are also discussed.

Re-evaluating efficiency of first-order numerical schemes for two-layer shallow water systems by considering different eigenvalue solutions

The efficiency of several first-order numerical schemes for two-layer shallow water equations (SWE) are evaluated here by considering different eigenvalue solutions. This study is a continuation of our previous work (Krvavica et al., 2018) in which we have proposed an efficient implementation of a Roe solver for two-layer SWE based on analytical expressions for eigenvalues and eigenvectors. In this work, the accuracy and computational cost of numerical, analytical, and approximated eigenvalue solvers are compared when implemented in Roe, Intermediate Field CaPturing (IFCP) and Polynomial Viscosity Matrix (PVM) schemes. Several numerical tests are performed to examine the overall efficiency of numerical schemes with different eigenvalue solvers when computing two-layer shallow-water flows. The results confirm that analytical eigenvalue solutions are much faster than numerical solvers, with a computational cost closer to approximate expressions. Consequently, the Roe scheme with analytical solutions to the eigenstructure is equally efficient as the IFCP scheme. On the other hand, IFCP and PVM schemes with analytical solutions to eigenvalues are found to be equally efficient as those with approximated expressions. Analytical eigenvalues show slightly better results when dealing with larger density differences between the layers.