Local Nonlinearity Research Articles

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

We present a brief overview of the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. First, we focus on the NLSE (both in the subcritical and the critical case) and, then, on the NLDE highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit by the convergence of the bound states.

Duffing Nonlinearity Localization via Extension Energy Confinement in an Elastic Mode Semicircular Beams Resonator

This letter reports a technique to suppress Duffing nonlinearity in silicon micromechanical resonators. A model of nonlinearity localization is established to analyze the confinement of nonlinear extension energy via vibration mode reconfiguration in beams resonators. Through a process of synchronous vibration between curved beams and its suspension beams, the vibration of resonator transits from stretch to elastic modes with the increase of the beam's curvature. Based on this principle, a microsemicircular beams resonator with double free-ends is introduced. Working on the elastic mode, the semicircular beams resonator shows a weakened instability of amplitude-frequency effect of only 1.3% of that of a conventional double-ended tuning forks resonator. With localized nonlinearity, the semicircular beams resonator can be operated at large vibration amplitude, thus achieving high-power handling performance with good frequency stability.

About Distributed Internal and Surface Strain Measurements within Prestressed Concrete Truck Scale Platforms

Distributed optical fibre measurement technology provides new possibilities in structural technical condition assessment in comparison with traditional spot measurements. It is not only possible to analyse strains continuously over structural member length with spatial resolution starting from as fine as 5 mm, but also the cracks and displacements state within reinforced concrete structures. The article presents pilot studies regarding prestressed concrete track scale platforms with a length of 8 to 16 m. Optical fibres were glued within composite rods attached to the stirrups inside the cross section of the slab on the height of lower and upper prestressing tendons, as well as they were glued to the concrete surface after bonding. That allows for comprehensive analysis of concrete strains state including all local nonlinearities (cracks) starting from hydration process (thermal-shrinkage strains), through prestressing tendons activation (strains regarding the transfer of compression forces from the tendond to the concrete) and finally during laboratory tests, when slabs were mechanically loaded until destruction. The ways of installation and exemplary results from selected phases of the research were presented as well as data interpretation was described and discussed.

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the $L^2$-subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit.

Local null controllability for a parabolic-elliptic system with local and nonlocal nonlinearities

Local null controllability for a parabolic-elliptic system with local and nonlocal nonlinearities

Free‐Decay Nonlinear System Identification via Mass‐Change Scheme

Methods for nonlinear system identification of structures generally require input‐output measured data to estimate the nonlinear model, as a consequence of the noninvariance of the FRFs in nonlinear systems. However, providing a continuous forcing input to the structure may be difficult or impracticable in some situations, while it may be much easier to only measure the output. This paper deals with the identification of nonlinear mechanical vibrations using output‐only free‐decay data. The presented method is based on the nonlinear subspace identification (NSI) technique combined with a mass‐change scheme, in order to extract both the nonlinear state‐space model and the underlying linear system. The technique is tested first on a numerical nonlinear system and subsequently on experimental measurements of a multi‐degree‐of‐freedom system comprising a localized nonlinearity.

Modeling and optimization of multi-item multi-constrained EOQ model for growing items

In this paper, a new mathematical model is presented, for the first time, for multi-item economic order quantity for growing items considering various operational constraints. The model aims to maximize company’s total profit by determining the optimal values of the decision variables including: number of ordered items from each type and the time period needed to grow each type of items. To propose a realistic model which can be implemented in variety of real-world problems, different operational constraints are considered including: on-hand budget, warehouse capacity, and total allowable holding cost constraints. To solve the problem in small sizes a well-known exact solution methodology called Sequential Quadratic Programming is utilized. To solve the problem in medium and large sizes two novel hybrid metaheuristics called Sine Cosine Crow Search Algorithm and Water Cycle Moth–Flame Optimization algorithm are utilized due to sub-optimal results of the exact solution method. This is due to existence of enormous number of local optima and non-linearity of the proposed model. The performance of the algorithms is evaluated using different measures such as relative percentage deviation, percentage relative error, and CPU-Time. By solving various test problems in different sizes performance of the algorithms is evaluated and the best solution method for the problem is determined using statistical analyses.

Mixed Local-Nonlocal Vector Schrödinger Equations and Their Breather Solutions**Supported by the Natural Science Foundation of Liaoning Province of China under Grant No. 201602678

To the best of our knowledge, all nonlinearities in the known nonlinear integrable systems are either local or nonlocal. A natural problem is whether there exist some nonlinear integrable systems with both local and nonlocal nonlinearities, and how to solve this kinds of spectral nonlinear integrable systems with both local and nonlocal nonlinearities. Recently, some novel mixed local-nonlocal vector Schrödinger equations are presented, which are different from the single local and nonlocal coupled Schrödinger equation. We investigate the Darboux transformation of mixed local-nonlocal vector Schrödinger equations with a spectral problem. Starting from a special Lax pairs, the mixed local-nonlocal vector Schrödinger equations are constructed. We obtain the one- and two- and N-soliton solution formulas of the mixed local-nonlocal vector Schrödinger equations with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-solitons are exhibited, the overtaking elastic interactions among the two-breather solitons are considered. We find that unlike the local and nonlocal cases, the mixed local-nonlocal vector Schrödinger equations have some novel results. The results in this paper might be helpful for understanding some physical phenomena described in plasmas.

Numerical Assessment of Reduced Order Modeling Techniques for Dynamic Analysis of Jointed Structures With Contact Nonlinearities

This work presents an assessment of classical and state of the art reduced order modeling (ROM) techniques to enhance the computational efficiency for dynamic analysis of jointed structures with local contact nonlinearities. These ROM methods include classical free interface method (Rubin method, MacNeal method), fixed interface method Craig-Bampton (CB), Dual Craig-Bampton (DCB) method and also recently developed joint interface mode (JIM) and trial vector derivative (TVD) approaches. A finite element (FE) jointed beam model is considered as the test case taking into account two different setups: one with a linearized spring joint and the other with a nonlinear macroslip contact friction joint. Using these ROM techniques, the accuracy of dynamic behaviors and their computational expense are compared separately. We also studied the effect of excitation levels, joint region size, and number of modes on the performance of these ROM methods.

Estimating Nonlinearities in p-Linear Hyperspectral Mixtures

Accurately estimating the elements in Earth observations is crucial when assessing specific features such as air quality index, water pollution, or urbanization process behavior. Moreover, physical–chemical composition can be retrieved from hyperspectral images when proper spectral unmixing architectures are employed. Specifically, when linear and nonlinear combinations of endmembers (pure spectral components) are accurately characterized, hyperspectral unmixing plays a key role in understanding and quantifying phenomena occurring over the instantaneous field-of-view. Thus, reliable detection of nonlinear reflectance behavior can play a key role in enhancing hyperspectral unmixing performance. In this paper, two new methods for adaptive design of mixture models for hyperspectral unmixing are introduced. One of the methods relies on exploiting geometrical features of hyperspectral signatures in terms of nonorthogonal projections onto the space induced by the endmembers’ spectra. Then, an iterative process aims at understanding the order of local nonlinearity that is displayed by each endmember over every pixel. An improved version of an artificial neural network-based approach for nonlinearity order information is also considered and compared. Experimental results show that the proposed approaches are actually able to retrieve thorough information on the nature of the nonlinear effects over the image, while providing excellent performance in reconstructing the given data sets.

Global and Multiplexed Dendritic Computations under In Vivo-like Conditions

SummaryDendrites integrate inputs nonlinearly, but it is unclear how these nonlinearities contribute to the overall input-output transformation of single neurons. We developed statistically principled methods using a hierarchical cascade of linear-nonlinear subunits (hLN) to model the dynamically evolving somatic response of neurons receiving complex, in vivo-like spatiotemporal synaptic input patterns. We used the hLN to predict the somatic membrane potential of an in vivo-validated detailed biophysical model of a L2/3 pyramidal cell. Linear input integration with a single global dendritic nonlinearity achieved above 90% prediction accuracy. A novel hLN motif, input multiplexing into parallel processing channels, could improve predictions as much as conventionally used additional layers of local nonlinearities. We obtained similar results in two other cell types. This approach provides a data-driven characterization of a key component of cortical circuit computations: the input-output transformation of neurons during in vivo-like conditions.

Mechanical characteristics of the bistable origami hypar

This letter explores the mechanical behavior of the origami hyperbolic paraboloid or hypar which is a bistable thin sheet structure with symmetric stable states. We use experiments to evaluate the global and local behaviors of the structure, which are unique from other bistable origami systems. In theory, for the hypar to reconfigure between the stable states, the thin sheet which is initially patterned with acute and reflex folds needs to fully flatten. However, we find that axial stretching as well as two distinctly different buckling patterns in the origami can allow for portions of the sheet to remain folded during the reconfiguration. The spacing of folds in the hypar, the resulting buckling characteristics, the stretching of the sheet, and the local material nonlinearity at the fold lines all have significant effects on the force–displacement response of the system. The work gives insight to these phenomena which are often omitted in the testing and modeling of more conventional origami-inspired systems. We reinforce our understanding of the hypar behavior by establishing an analytical model that can simulate the kinematics, the buckling, and the force–displacement behavior of the structure. The letter also discusses topics for future extension and introduces hypar chains where multiple hypars are connected in series to achieve multistability.

Efficient computational approach for fatigue assessment of riveted connections

This paper presents an innovative computational approach for fatigue assessment of riveted connections. Such methodology is based on modal superposition principles, submodelling techniques and an elastoplastic post-processing to efficiently evaluate fatigue damaging events considering local geometrical, material and contact nonlinearities. The computational burden, intrinsic to localized detailed numerical analyses, is mitigated implementing a sub-algorithm based on network parallel computing. The developed computational methodology overcomes relevant drawbacks of S–N global approaches suggested in international standards and guidelines and it is an accurate methodology to analyse complex riveted fatigue-critical details, in particular the ones part of large structures, e.g. railway bridges.An idealised riveted beam case-study was investigated and local damage parameters were calculated in reduced computational times. The accuracy of such results proved that the proposed approach allows fully analysing the fatigue life of riveted connections, being applicable to unlimited geometrical configurations submitted to variable loadings.

Vibration analysis of beam structures with localized nonlinearities by a wave approach

This paper presents a general wave propagation approach for the vibration analysis of structures consisting of beam elements with localized nonlinearities. The scattering of a wave at a nonlinear discontinuity creates multi-harmonic waves. This forms the basis of the model formulation in which the displacement in each element composing the structure is defined as a sum of waves associated with the fundamental frequency and its harmonics. The coupling between separate elements is expressed using transmission coefficients function of the harmonic number and includes propagating and decaying near field waves whose amplitudes are the unknowns of the problem. With the use of harmonic balance, a characteristic algebraic equation of motion is derived. The general systematic methodology is demonstrated and detailed on beam-like systems with nonlinear boundaries for validation against a conventional finite element analysis. To illustrate the effectiveness of the approach on a more practical case, the method is also applied to a simplified model of a turbomachinery blade with underplatform friction dampers. The predictions of stick-slip effect and global nonlinear damping behavior made using the proposed approach are shown to be in excellent agreement with the results of numerical time integration with the advantage of reduced computational costs.

Detection and Location of Nonlinearities using Reciprocity Breakdown

Damage in a structure often results in local stiffness nonlinearities and detecting these nonlinearities can be used to monitor the health of the structure. It is well-known that nonlinearities in structures lead to a breakdown in reciprocity, where the frequency response function between two points on the structure depends upon the forcing location. This paper proposes a measure to quantify the level of non-reciprocity in a structure and investigates the effect of the location and form of nonlinearity on this non-reciprocity measure. A simulated discrete mass-spring system was used to determine the effect of the excitation and response locations on the ability to detect the nonlinearity. Stepped-sine testing is commonly used to characterise a nonlinear system since harmonic excitation emphasises nonlinear phenomena and can, for example, allow the system to exhibit multiple solutions. Thus, a simulation of a stepped sine test was used as the benchmark to highlight reciprocity breakdown in the most favourable case. However, impact excitation is much easier and faster to implement in practice and consequently the effect of the type of excitation on the detection and location of nonlinearities was considered. Finally, the prospects for using a measure of reciprocity in a structural health monitoring system are discussed.

A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It relies on a domain decomposition method which introduces several subdomains of interest (called patches) containing the different sources of uncertainties and non-linearities. An iterative algorithm is then introduced, which requires the solution of a sequence of linear global problems (with deterministic operators and uncertain right-hand sides), and non-linear local problems (with uncertain operators and/or right-hand sides) over the patches. Non-linear local problems are solved using an adaptive sampling-based least-squares method for the construction of sparse polynomial approximations of local solutions as functions of the random parameters. Consistency, convergence and robustness of the algorithm are proved under general assumptions on the semi-linear elliptic operator. A convergence acceleration technique (Aitken’s dynamic relaxation) is also introduced to speed up the convergence of the algorithm. The performances of the proposed method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction problem.

Rescaling Approach for a Stochastic Population Dynamics Equation Perturbed by a Linear Multiplicative Gaussian Noise

We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat $O,$ governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on $\partial O$ with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear multiplicative Gaussian noise perturbation in the equation. The main result proves that the stochastic model is well-posed, the solution being in the class of path-wise continuous functions and satisfying some particular regularities with respect to the age and space. The approach is based on a rescaling transformation of the stochastic equation into a random deterministic time dependent hyperbolic-parabolic equation with local Lipschitz nonlinearities. The existence and uniqueness of a strong solution to the random deterministic equation is proved by combined semigroup, variational and approximation techniques. The information given by these results is transported back via the rescaling transformation towards the stochastic equation and enables the proof of its well-posedness.

Inelastic response analysis of bridges subjected to non-stationary seismic excitations by efficient MCS based on explicit time-domain method

This paper is devoted to stochastic response analysis of large-scale bridges with local plastic deformations under non-stationary random seismic excitations. A force-based fiber beam–column element is employed to simulate the inelastic behavior of bridges. By use of the explicit time-domain expressions of seismic responses, an effective dimension-reduced method is developed for fast inelastic time-history analysis of a structure with local nonlinearities. Only the nodal displacements that are associated with the inelastic elements need to be solved through iteration process, which leads to a significant reduction in computational cost for inelastic time-history analysis. With the merit of high efficiency, the dimension-reduced method is further used as a computational tool for repetitive inelastic time-history analyses in Monte Carlo simulation, and the statistical moments and mean peak values of critical responses can be readily achieved under random seismic excitations. To demonstrate the feasibility of the present approach, the nonlinear random vibration analysis of a long-span suspension bridge with a main span of 1200 m is carried out under rare earthquakes, in which the inelastic behaviors of the main towers are investigated with the use of inelastic fiber elements.

Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases

We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:{−ε2Δu+V(x)u+λ1|u|2u+λ2(K⁎|u|2)u=0 in R3,u>0, u∈H1(R3), where ε is a small positive parameter, λ1,λ2∈R, ⁎ denotes the convolution, K(x)=1−3cos2⁡θ|x|3 and θ=θ(x) is the angle between the dipole axis determined by (0,0,1) and the vector x. Under certain assumptions on (λ1,λ2)∈R2, we construct a family of positive solutions uε∈H1(R3) which concentrates around the local minima of V as ε→0. Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation.

Ground state solutions for the Hénon prescribed mean curvature equation

Abstract In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in {{\mathbb{R}^{N}}} , both in the Euclidean and in the Minkowski spaces. Motivated by the studies of Ni and Serrin [W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Convegni Lincei 77 1985, 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state. We also present a partial result on the existence of a ground state solution in the Minkowski space.