Nonlinear Eigenvalue Equations Research Articles

Monte Carlo Calculation Method for Reactor Period Utilizing the Differential Operator Sampling Technique

The inverse reactor period α is a fundamental mode eigenvalue of the α-mode nonlinear Boltzmann eigenvalue equation that considers delayed neutron contributions. Thus far, several Monte Carlo methods, including the α-k, weight balancing, and transition rate matrix methods, have been developed to calculate α. This study presents a new Monte Carlo method for predicting α by using the derivatives of the k-eigenvalue with respect to α. Formulae are derived to calculate the first and second derivatives using the differential operator sampling method. The key feature of the new proposed method is its ability to estimate the uncertainty of the predicted α by considering the uncertainty of the k-eigenvalue and its derivatives with respect to α.

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

We consider the non-linear eigenvalue equations characterizing Lp into Lq Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case (p,q)∈(1,∞)2. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if 1p+1q=1. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

An overview of a posteriori error estimation and post-processing methods for nonlinear eigenvalue problems

In this article, we present an overview of different a posteriori error analysis and post-processing methods proposed in the context of nonlinear eigenvalue problems, e.g. arising in electronic structure calculations for the calculation of the ground state and compare them. We provide two equivalent error reconstructions based either on a second-order Taylor expansion of the minimized energy, or a first-order expansion of the nonlinear eigenvalue equation. We then show how several a posteriori error estimations as well as post-processing methods can be formulated as specific applications of the derived reconstructed errors, and we compare their range of applicability as well as numerical cost and precision.

The Standard Model and quantum state reduction from Heim’s field theory

Abstract Core parts of the Standard Model are derived from B. Heim’s quantum field theory, whose poly-metric describes spacetime and matter in a unified formalism. Its non-linear eigenvalue equation transforms into the Einstein field equation in the macroscopic limit. The 6-dimensional Heim space can be determined as locally isomorphic to a SU(2) ⊗ SU(2) ⊗ U(1) ⊗ U(1) symmetry and thus to the SU(3), which allows to connect to the local gauge symmetries and boson fields of the Standard Model. The Fermion and Higgs field and their coupling are deduced from Heim’s basic equations, providing new insight into possible correlations of these fields. Furthermore, the derivation yields an additional imaginary coupling term which seems to account for quantum mechanical state reduction in the non-relativistic limit. The recently performed calculation of the mass spectrum of elementary particles in a new approach based on Heim’s theory (with average error to the data < 1 % ${< } 1\%$ ) appears as even more relevant, having now shown that the theory can connect to the achievements of the Standard Model.

The heterogeneous helicoseir

We study the rotations of a heavy string (helicoseir) about a vertical axis with one free endpoint and with arbitrary density, under the action of the gravitational force. We show that the problem can be transformed into a nonlinear eigenvalue equation, as in the uniform case. The eigenmodes of this equation represent equilibrium configurations of the rotating string in which the shape of the string does not change with time. As previously proved by Kolodner for the homogeneous case, the occurrence of new modes of the nonlinear equation is tied to the spectrum of the corresponding linear equation. We have been able to generalize this result to a class of densities ρ(s)=γ(1−s)γ−1, which includes the homogeneous string as a special case (γ=1).We also show that the solutions to the nonlinear eigenvalue equation (NLE) for an arbitrary density are orthogonal and that a solution of this equation with a given number of nodes contains solutions of a different helicoseir, with a smaller number of nodes. Both properties hold also for the homogeneous case and had not been established before.

Numerical hygrothermal frequency of pre-damage shallow shell panel: A nonlinear FE approach

ABSTRACT In this work, the nonlinear eigenvalues of the debonded composite panel (flat, single and doubly curve) are evaluated under unlike humidity and temperature conditions. To obtain the said modal responses a macro mechanical isoparametric finite element-based numerical model of pre-damage laminated composite is developed. The structural panel deformations are counted through two different kinematic theories with small deviations (related to the drilling degrees of freedom). The geometrical distortions of the panel component are expressed using the common Green’s strain relation in Lagrangian scope. The governing differential equation (converted to nonlinear eigenvalue equations) is derived from a dynamic version of the variational principle. The direct iterative approach has been adopted to compute the nonlinear frequencies in association with the finite element steps. Also, the element sensitivity behavior of the obtained numerical solution is verified and extended further to check the efficacy (comparing the results from the published domain). In continuation, the nonlinear modal values are computed through a series of examples considering the role of debonding with the structural parameters in detail.

The generation of mass in a non-linear field theory

Abstract The mass spectrum of elementary particles is calculated in a new approach, based on B. Heim’s quantum field theory, which manifests in a non-linear eigenvalue equation and merges into the Einstein field equation in the macroscopic limit. The poly-metric of the theory allows spacetime and matter to be described in a unified formalism, representing a radical geometrisation of physics. The calculated mass energies are in very good agreement with the empirical data (error < 1 % ${< }1\%$ on average) if the mass scale is gauged to the electron as lowest mass and the second main parameter, determining the strength of obtained mass hierarchy levels, is close to the half inverse of the fine structure constant, describing the difference in strength between the electromagnetic and the strong interaction. The obtained hierarchy levels are not identical to the particle generations of the Standard Model; however, show a self-similarity typical for non-linear theories. For higher values of the main quantum number N, the calculated mass formula becomes identical to the phenomenological formulae of Nambu, respectively, Mac Gregor.

Thermal Post-Buckling Strength Prediction and Improvement of Shape Memory Alloy Bonded Carbon Nanotube-Reinforced Shallow Shell Panel: A Nonlinear Finite Element Micromechanical Approach

Abstract This article reported first-time the post-buckling temperature load parameter values of nanotube-reinforced polymeric composite panel and their improvement by introducing the functional material (shape memory alloy, SMA) fiber. The temperature load values of nanotube composite and SMA activation are modeled using the single-layer type higher-order kinematic model in association with isoparametric finite element technique. To ensure the effective properties of SMA bonded nanotube composite under the elevated temperature, a hybrid micromechanical material modeling approach is adopted (Mori–Tanaka scheme and rule of mixture). The present structural geometry distortion under elevated temperature is modeled through the nonlinear strain kinematics (Green–Lagrange), whereas the strain reversal achieved with the help of marching technique (inclusion of material nonlinearity). Owing to the importance of geometrical distortion of the polymeric structure, the current model includes all of the nonlinear strain terms to accomplish the exact deformation. Further, to compute the post-buckling responses, the governing nonlinear eigenvalue equations are derived by Hamilton's principle. The numerical solution accuracy is verified with adequate confirmation of model consistency. The material model applicability for different structural configurations including important individual/combined parameter tested through a series of examples. Moreover, the final understanding relevant to the post-buckling characteristics of the polymeric structure and SMA influences is highlighted in details considering the prestrain, recovery stress, and their volume fractions.

Acoustic modes of locally resonant phononic crystals: Comparison with frequency-dependent mass models

Locally resonant acoustic materials were proposed to control sound using structures with feature size orders of magnitude smaller than the acoustic wavelength in air. We analytically derive the effective, frequency-dependent mass densities of resonant oscillators consisting of a heavy mass within a light shell, embedded in foam, in two dimensions, using a rigid core-shell approximation. The effective mass is expressible in closed form by elementary functions, enabling a mapping of the low-frequency physics to a simple one-dimensional model involving a point mass harmonically coupled within a box. The acoustic band structure of a two-dimensional square lattice of these oscillators is evaluated in the effective, frequency-dependent mass density model and compared with the exact solution. For the out-of-plane oscillation within this two-dimensional phononic crystal, the nonlinear eigenvalue equation resulting from the frequency-dependent density is solved with the Cutting Surface Method to yield the acoustic band structure. This agrees with the exact band structure obtained by finite element method calculations within $2%$ numerical error. For the in-plane oscillations involving internal translations or rotations within the core-shell resonators, the one-dimensional mass-in-a-box model recaptures the exact acoustic modes at only certain high-symmetry points of the two-dimensional Brillouin zone. The description of more complex resonances within acoustic metamaterials requires a fundamental generalization of current effective mass models.

A hyper-block self-consistent approach to nonlinear Schrodinger equations: Breeding, metamorphosis, and killing of Hofstadter Butterflies

Nonlinear Schrödinger equations play essential roles in different physics and engineering fields. In this paper, a hyper-block finite-difference self-consistent method (HFDSCF) is employed to solve this stationary nonlinear eigenvalue equation and demonstrated its accuracy. By comparing the results with the Sinc self-consistent (SSCF) method and the exact available results, we show that the HFDSCF gives quantum states with high accuracy and can even solve the strongly nonlinear Schrodinger equations. Then, by applying our method to the Hofstadter butterfly problem, we describe the breeding, metamorphosis, and killing of these butterflies by using nonlinear interactions and two constant length multi-well and sinusoidal potentials.

Nonlinear vibration analysis of third-order shear deformable functionally graded beams by a new method based on direct numerical integration technique

This study investigates the nonlinear vibration and dynamic response of a beam made of functionally graded material (FGM) within the framework of the improved third-order shear deformable theory. The beam is subjected to a concentrated moving load, and the included angle of the load direction and axial direction varies with time. Considering the von Karman geometric nonlinearity, the nonlinear formulations of the FGM beam is derived. The Newmark method in conjunction with the Newton–Raphson iteration is adopted to analyze the dynamic response of the beam. A new method, based on a direct numerical integration technique for the matrix form motion equation, is presented to study the nonlinear vibration of the FGM beam. The proposed method overcomes the problem of signal determination for the solution of the eigenvector equation that often leads to nonconvergence or a false result of the nonlinear eigenvalue equation. In relation to the numerical results, the effects of material property distribution, vibration amplitude on the nonlinear dynamic behavior of the FGM beams, and the energy transference phenomenon are discussed in this paper.

A Cascadic Adaptive Finite Element Method for Nonlinear Eigenvalue Problems in Quantum Physics

This paper introduces a cascadic adaptive finite element method for nonlinear eigenvalue equations arising from quantum physics following the multilevel correction strategy. Instead of the classical scheme, which requires solving nonlinear eigenvalue equations on a series of adaptive spaces directly, the new scheme consists of several smoothing processes on an adaptive space sequence and nonlinear eigenvalue equations being solved in a very low dimensional space. The main feature of the proposed scheme is that large-scale nonlinear eigenvalue problem solving is avoided, and the associated smoothing process can be executed efficiently based on the appropriate number of smoothing steps. Thus, efficiency can be enhanced by the proposed cascadic adaptive method. The good performance of the new finite element strategy is examined by various numerical experiments.

Vibrations of graphene nanoplatelet reinforced functionally gradient piezoelectric composite microplate based on nonlocal theory

This paper investigates the small-scale effect on the linear and nonlinear vibrations of the graphene nanoplatelet (GNPL) reinforced functionally gradient piezoelectric composite microplate based on the nonlocal constitutive relation and von Karman geometric nonlinearity. The GNPL reinforced functionally gradient piezoelectric composite microplate is resting on the Winkler elastic foundation and is subjected to an external electric potential. The parallel model of Halpin Tsai is used to compute the effective Young’s modulus of the GNPL reinforced functionally gradient piezoelectric composite microplate. The Poisson’s ratio, mass density and piezoelectric properties of the GNPL reinforced functionally gradient piezoelectric composite microplate are calculated by using the rule of mixture. Hamilton’s principle is adopted to obtain the higher-order nonlinear partial differential governing equations of motion for the GNPL reinforced functionally gradient piezoelectric composite microplate. The partial differential governing equations of motion are reduced to a system of the nonlinear algebraic eigenvalue equations by using the differential quadrature (DQ) method and are solved by an iteration progress. The efficiency and accuracy of the present approach are verified by comparing with the existed results. Both uniformly and functionally distributing graphene nanoplatelets (GNPLs) are considered to investigate the effects of the GNPL concentration, external voltage, nonlocal parameter, geometrical and piezoelectric characteristics of the GNPLs as well as the elasticity coefficient of the Winkler elastic foundation on the linear and nonlinear dynamic behaviors of the GNPL reinforced functionally gradient piezoelectric composite microplate with various boundary conditions. The numerical results clearly manifest that the GNPLs can significantly enhance the structural stiffness of the micro-electro-mechanical system (MEMS).

Semi-Supervised Domain Adaptation by Covariance Matching.

Transferring knowledge from a source domain to a target domain by domain adaptation has been an interesting and challenging problem in many machine learning applications. The key problem is how to match the data distributions of the two heterogeneous domains in a proper way such that they can be treated indifferently for learning. We propose a covariance matching approach DACoM for semi-supervised domain adaptation. The DACoM embeds the original samples into a common latent space linearly such that the covariance mismatch of the two mapped distributions is minimized, and the local geometric structure and discriminative information are preserved simultaneously. The KKT conditions of DACoM model are given as a nonlinear eigenvalue equation. We show that the KKT conditions could at least ensure local optimality. An efficient eigen-updating algorithm is then given for solving the nonlinear eigenvalue problem, whose convergence is guaranteed conditionally. To deal with the case when homogeneous information could only be matched nonlinearly, a kernel version of DACoM is further considered. We also analyze the generalization bound for our domain adaptation approaches. Numerical experiments on simulation datasets and real-world applications are given to comprehensively demonstrate the effectiveness and efficiency of the proposed approach. The experiments show that our method outperforms other existing methods for both homogeneous and heterogeneous domain adaptation.

Thermal buckling of functionally graded triangular microplates

The thermal buckling behavior of thin to moderately thick functionally graded isosceles triangular microplates with temperature-dependent material properties is investigated. The governing equations are derived based on the modified strain gradient theory (MSGT) in conjunction with the first-order shear deformation theory. The adjacent equilibrium criterion and Chebyshev–Ritz method are employed to derive the nonlinear thermal buckling eigenvalue equations, which are solved by a direct iterative method. The fast rate of convergence and accuracy of the method are demonstrated numerically. Then, the effects of length scale parameters, material gradient index, different boundary conditions, apex angle and ratio of width to thickness on the critical temperature rises of the triangular microplates are studied. In addition, comparisons between the results of MSGT and modified couple stress theory and classical theory (CT) are performed. The results show that by increasing the apex angle, the critical temperature rise increases, but increase in the material gradient index and the dimensionless length scale parameter decreases the critical temperature rise. In addition, it is observed that by considering the temperature dependence of material properties, the critical temperature rises decrease significantly. Also, the MSGT and CT yield the highest and the lowest critical temperature rise, respectively.

Linear and nonlinear free and forced vibrations of graphene reinforced piezoelectric composite plate under external voltage excitation

Graphene reinforcements can obviously enhance the piezoelectric properties as well as the mechanical properties of the polyvinylidene fluoride (PVDF). This paper investigates the linear and nonlinear vibration behaviors of the smart piezoelectric composite plate reinforced by uniformly and non-uniformly dispersing graphene platelets (GPLs). The effective Young’s modulus is predicted by the Halpin Tsai’s parallel model while the effective mass density, Possion’s ratio and piezoelectric properties are calculated by the rule of the mixture. Based on the first-order shear deformation plate theory, von Karman nonlinear geometric relationship and Hamilton’s principle, the governing equations of motion under different boundary conditions are derived for the smart piezoelectric composite plate. The governing equations of motion are solved to obtain the nonlinear eigenvalue equations by the differential quadrature (DQ) method. The analysis is validated by comparing with the current results of the smart piezoelectric composite plate. The effects of the GPL distribution pattern, stratification number, concentration and geometry of GPLs, plate geometry, external voltage and piezoelectric properties of GPLs as well as boundary conditions on the linear and nonlinear vibration behaviors are discussed in detail. The numerical results clearly illustrate that there exists the great potential for using GPLs in achieving smart structures with significantly improved structural stiffness.

Thermal post-buckling of temperature dependent sandwich plates with FG-CNTRC face sheets

ABSTRACTPresent research investigates the thermal postbuckling of sandwich plates containing a stiff core and two thin carbon nanotube reinforced composite (CNTRC) face sheets. Properties of the core, carbon nanotubes (CNTs) and polymeric matrix of the faces are assumed to be temperature-dependent. It is assumed that CNTs as reinforcements may be distributed according to a functionally graded pattern. Plate is formulated based on the first-order shear deformation theory and von Kármán type of geometrical nonlinearity. The governing equations are obtained by the energy method with the aid of the Conventional Ritz method. Shape functions of the Ritz method are estimated according to the Chebyshev polynomials. A set of nonlinear eigenvalue equations is achieved. The obtained equations are homogeneous, coupled, and nonlinear in terms of both displacements and temperature. A successive displacement control strategy is implemented to trace the thermal postbuckling equilibrium path of the plate. It is shown that, with increasing the volume fraction of CNT, critical buckling temperature of sandwich plate increases and postbuckling deflection decreases. Furthermore, through a functionally graded distribution of volume fraction of CNTs across the thickness, critical buckling temperature of the sandwich plate may be enhanced and thermal postbuckling deflection may be alleviated.

Thermal buckling and postbuckling analysis of size-dependent composite laminated microbeams based on a new modified couple stress theory

Based on a new modified couple stress theory for composite laminates, considering geometric nonlinear theory and Timoshenko beam hypothesis, the governing equations for size-dependent composite laminated microbeams in thermal environment are derived using Hamilton’s principle. Analytical and numerical solutions are employed in solving the present problem, respectively. An auxiliary function is introduced to reduce the governing equations to a single fourth-order integral-differential equation, and the exact solutions for the thermal buckling and postbuckling of microbeams with combination of in-plane immovable simply supported boundary conditions are obtained. By introducing the differential quadrature method, the governing equations are transferred into a system of nonlinear algebraic eigenvalue equations. In numerical examples, comparison between the present results and those obtained in the literature verifies the validity and efficiency of the present analytical and numerical methods. The effects of thermal expansion coefficients and material length scale parameter are discussed. Numerical results indicate that the above-mentioned effects play very important roles for the thermal buckling and postbuckling of the composite laminated microbeams.

Thermoelastic buckling analysis of pre-twisted functionally graded beams with temperature-dependent material properties

As a first endeavor, the thermal buckling behavior of pre-twisted functionally graded (FG) beams with temperature-dependent material properties is investigated. The governing stability equations are derived based on the third-order shear deformation theory (TSDT) in conjunction with the adjacent equilibrium state criterion under the von Kármán's nonlinear kinematic assumptions using the Chebyshev-Ritz method. The Chebyshev polynomials multiplied with some suitable boundary functions are used as the basis functions, which allow one to analyze the beams with different boundary conditions. The extracted system of nonlinear algebraic eigenvalue equations is solved iteratively to obtain the critical temperature rise. The convergence behavior together with accuracy of the solution method and the correctness of formulation are demonstrated through different examples. Then, the influences of the linear and nonlinear variation of the angle of twist along the beam axis, the value of twist angle, length-to-thickness ratio, thickness-to-width ratio, material gradient index and temperature dependence of material properties on the critical temperature rise of the pre-twisted FG beams under different boundary conditions are investigated. It is shown that the pre-twist angle increases the thermal buckling resistance of the pre-twisted FG beams, but the temperature dependence of material properties reduces it.

Thermal buckling of rotating pre-twisted functionally graded microbeams with temperature-dependent material properties

As a first endeavor, the thermal buckling of rotating pre-twisted functionally graded (FG) microbeams with temperature-dependent material properties is studied based on the modified strain gradient theory in conjunction with the first-order shear deformation theory of beams. The adjacent equilibrium criterion and Chebyshev–Ritz method are employed to derive the nonlinear algebraic eigenvalue equations governing the thermal buckling behavior of the microbeams, which are solved iteratively. The fast rate of convergence and accuracy of the method are numerically demonstrated. Then, the effects of the twist angle, rate of twist angle (as an important geometrical design parameter), material length scale parameter, material gradient index and angular velocity on the thermal load-bearing capacity of rotating pre-twisted FG microbeams under different boundary conditions are studied. It is shown that by increasing the hub radius, the angular velocity and the length scale parameter, the thermal buckling load increases, but an increase of the material gradient index reduces the critical thermal buckling load. In addition, the formulation can be easily degenerated to those of large-scale rotating pre-twisted FG beams.