Local Nonlinearity Research Articles

Nonlinear substructuring in the modal domain: numerical validation and experimental verification in presence of localized nonlinearities

In many systems of interest, most of the structure is well approximated as linear but some parts must be treated as nonlinear to get accurate response predictions: significant nonlinear effects are due to the connections between coupled subsystems, such as in automotive or aerospace structures. The present work aims at predicting the nonlinear behavior of coupled systems using a substructuring technique in the modal domain. This study focuses on the effects of nonlinear connections on the dynamics of an assembly in which the coupled subsystems can be considered as linear. Each connection is instead considered as a quasi-linear substructure with stiffness that is function of amplitude or energy. The iterative procedure used here is enhanced with respect to previous works by enforcing a better control of the total energy at each iteration allowing to obtain the solution for a prescribed set of energy levels. Also, the initial guess and the convergence criterion have been modified to speed up the procedure. This technique is applied to a system made of two continuous linear subsystems coupled by nonlinear connections. The numerical results of the coupling are first compared to the ones obtained by using the Harmonic Balance technique on the model of the complete assembly to evaluate its effectiveness and understand the effects of modal truncation. Besides, a nonlinear connecting element, specifically designed in order to have a nearly cubic hardening behavior, is used in an experimental setup. Substructuring results are compared to experimental results measured on the assembled system, in order to evaluate the correlation between mode shapes and the accuracy in the resonance frequency at several excitation levels.

Thermalization in the one-dimensional Salerno model lattice.

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

Numerical solution of additive manufacturing problems using a two‐level method

AbstractAdditive manufacturing techniques have grown significantly in recent years due to their ability to produce designs almost impossible to obtain with standard production technologies. This growth has in turn led to increased demand for numerical simulation of additive processes. Unfortunately, such simulations are difficult from the computational point of view, since additive problems are characterized by the presence of highly local nonlinearities requiring fine mesh resolutions on small portions of the domain. These difficulties are compounded by the fact that such regions of local high nonlinearities move in the domain with time, and further, by the fact that the physical domain itself evolves with time. Accordingly, the present article addresses these complex computational issues by employing a two‐level technique. The two‐level method provides a natural framework for the resolution of localized nonlinearities while avoiding the need for complex meshing operations, such as refinement‐and‐derefinement algorithms. The proposed approach is validated on a series of one‐ and two‐dimensional problems to establish the viability of the introduced two‐level method for additive manufacturing problems.

Kron's substructuring method to the calculation of structural responses and response sensitivities of nonlinear systems

Numerous civil engineering structures exhibit nonlinearities. Even though the nonlinearities constitute only a small part of the structure, the entire structure behaves nonlinearly, and the analysis of the whole structure is consequently computationally expensive. This paper develops Kron's substructuring method for fast computation of structural responses and response sensitivities of nonlinear systems. The nonlinearity is detected and located using the ordinary coherence function. The global structure is then divided into linear and nonlinear substructures. The local nonlinearities are thus restricted in a few nonlinear substructures. The linear substructural responses are interpreted as the combination of a few master modal responses based on the mode superposition. The discarded slave modal responses of the linear substructures are compensated by the corresponding master modal responses, nonlinear substructural responses and external excitation on the linear substructures. A reduced vibration equation of a much smaller size is then derived with the transformation matrices. The structural responses and response sensitivities are calculated from the reduced vibration equation. The precision and efficiency of the proposed method are verified by a nonlinear spring-mass system and a relatively large-scale nonlinear frame.

Direct optimisation based model selection and parameter estimation using time-domain data for identifying localised nonlinearities

A major difficulty when modelling nonlinear structures from experimental vibration data is to determine the type of nonlinear functions that will better predict their dynamic response. In this paper we address this issue by developing a recursive framework in which the characteristics and parameters of nonlinear structures are identified using measured input and output time-domain data. Forward-backward and exhaustive search regression algorithms are exploited based on optimisation techniques to recursively select and quantify the best nonlinear functions from a predefined library of nonlinear terms. The framework assumes localised nonlinearities for which their location is assumed to be known. The proposed methodology is demonstrated using numerical and experimental examples of single and multi-degree-of-freedom systems. The results presented highlight key advantages of the proposed method including: the capability of treating multi-degree of freedom nonlinear systems holding different types of localised nonlinearities, and the capability of selecting nonlinear terms with a light computational effort and with limited number of time samples.

Application of Direct Integration Methods in the solution of a nonlinear beam problem

This work applies different numerical methods involved in the solution of a nonlinear clamped beam problem. The methodology used in the discretization of the dynamic problem is based on the Finite Element Method (FEM), followed by mode superposition, where a localized nonlinearity is applied at the free end of the beam. The solution of the nonlinear problem is performed by five different integration methods. The solution code is implemented in FORTRAN language, validated with ANSYS and the dynamic response and the graphs are obtained with the help of MATLAB software. The work shows the convergence of the implemented methods with various validation problems.

Mixed finite elements for convection-coupled phase-change in enthalpy form: Open software verified and applied to 2D benchmarks

Melting and solidification processes are often affected by natural convection of the liquid, posing a multi-physics problem involving fluid flow, convective and diffusive heat transfer, and phase-change reactions. Enthalpy methods formulate this convection-coupled phase-change problem on a single computational domain. The governing equations can be solved accurately with a monolithic approach using mixed finite elements and Newton’s method. Previously, the monolithic approach has relied on adaptive mesh refinement to regularize local nonlinearities at phase interfaces. This contribution instead separates mesh refinement from nonlinear problem regularization and provides a continuation procedure which robustly obtains accurate solutions on the tested 2D uniform meshes. A flexible and extensible open source implementation is provided. The code is formally verified to accurately solve the governing equations in time and in 2D space, and convergence rates are shown. Two benchmark simulations are presented in detail with comparison to experimental data sets and corresponding results from the literature, one for the melting of octadecane and another for the freezing of water. Sensitivities to key numerical parameters are presented. For the case of freezing water, effective reduction of numerical errors from these key parameters is successfully demonstrated. Two more simulations are briefly presented, one for melting at a higher Rayleigh number and one for melting gallium.

Modal energy exchanges in an impulsively loaded beam with a geometrically nonlinear boundary condition: computation and experiment

The capability of a geometrically nonlinear boundary condition, i.e., a strong local nonlinearity, in “redistributing” a broadband input energy (generated by an impulsive load) among the vibration modes of a cantilever Euler–Bernoulli beam is investigated. It is shown that this modal energy redistribution increases the inherent capacity of the cantilever for passive energy dissipation. The nonlinear boundary condition is realized by grounding the free end of the cantilever through an inclined linear spring–damper pair with initial angle of inclination $$ \phi_{0} $$ relative to the neutral axis of the beam while at rest. For $$ \phi_{0} < 90^{^\circ } $$ , the inclined spring–damper pair is geometrically nonlinear, whereas in the limiting case $$ \phi_{0} = 90^{^\circ } $$ the boundary condition becomes linear. To study the nonlinear modal energy redistribution in the cantilever, a multi-step system identification method to identify the unknown parameters of the experimental fixture is employed; this informs a computational reduced-order finite element (FE) model of the fixture. First, the Multi-input Multi-output Frequency Domain Identification (MFDID) technique to analyze the experimental frequency response functions of the “base” linear cantilever without the boundary condition is employed and its modal parameters are identified. Next, the boundary condition for the limiting angle $$ \phi_{0} = 90^{^\circ } $$ is imposed, so that again a linear fixture is obtained. Through reconciliation of computational and experimental measurements, the (linear) stiffness and damping coefficients of the boundary are identified, as well. Finally, by varying the angle of inclination in the range $$ 0^\circ \le \phi_{0} < 90^{^\circ } $$ , the nonlinear transient responses of the identified FE model with the nonlinear boundary condition are computed and projected onto the linearized modal basis of the system in the limit of zero energy. The computational FE results favorably compare with experimental measurements. Following this, the time-averaged modal energies of the system are computed and used to estimate the portion of the total energy of the beam allocated to each mode. Additionally, by employing these modal energies one may study and track the nonlinear energy exchanges between subsets of modes for different angles of initial inclination $$ \phi_{0} $$ of the nonlinear boundary attachment. The computational results are validated by experimental measurements, thus highlighting the predictive capacity of the computational FE model. In the last step, a scalar measure for modal energy exchange is defined by computing the maximum fluctuation in the percentage of each of the instantaneous modal energies that is the maximum percentage of energy being exchanged by the modes. This measure proves to be dependent on both the initial energy and the initial angle of inclination $$ \phi_{0} $$ . Again, experimental measurements favorably compare to computational FE simulations.

Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

&lt;abstract&gt;&lt;p&gt;In this paper, by using the mountain pass lemma and the skill of truncation function, we investigate the existence and concentration phenomenon of nontrivial weak solutions for a class of elastic beam differential equation with two parameters $ \lambda $ and $ \mu $ when the nonlinear term satisfies some growth conditions only near the origin. In particular, we obtain a concrete lower bound of the parameter $ \lambda $, and analyze the relationship between $ \lambda $ and $ \mu $. In the end, we investigate the concentration phenomenon of solutions when $ \mu\to 0 $, and obtain a specific lower bound of the parameter $ \lambda $ which is independent of $ \mu $.&lt;/p&gt;&lt;/abstract&gt;

Non-differentiable solutions for non-linear local fractional heat conduction equation

Fractional calculus has many advantages. Under consideration of this paper is a (2+1)-dimensional non-linear local fractional heat conduction equation with arbitrary degree non-linearity. Backlund transformation of a reduced form of the local fractional heat conduction equation is constructed by Painleve analysis. Based on the Backlund transformation, some exact non-differentiable solutions of the local fractional heat conduction equation are obtained. To gain more insights of the obtained solutions, two solutions are constrained to a Cantor set and then two spatio-temporal fractal structures with profiles of these two solutions are shown. This paper further reveals by local fractional heat conduction equation that fractional calculus plays important role in dealing with non-differentiable problems.

A high-performance method of vibration analysis for large-scale systems with local strong nonlinearity (Dimension reduction method using new type of complex modal analysis)

A high-performance method of vibration analysis for large-scale systems with local strong nonlinearity (Dimension reduction method using new type of complex modal analysis)

Stability design criteria for closely spaced built-up stainless steel columns

This paper aims to develop design recommendations for closely spaced built-up stainless steel columns, based on findings gained in performed research at the University of Belgrade. The research focuses on pin-ended built-up columns formed from two press-braked channel chords oriented back-to-back and addresses their flexural buckling capacity about the minor axis. The impact of overall and local chord slenderness, interconnection stiffness, geometric imperfections and material nonlinearity is evaluated. In order to fully exploit their structural performance, two separate approaches for the design of built-up columns with welded or bolted interconnections are defined that include different formulas for shear stiffness.

An automatically generated high-resolution earthquake catalogue for the 2016–2017 Central Italy seismic sequence, includingPandSphase arrival times

SUMMARYThe 2016–2017 central Italy earthquake sequence began with the first main shock near the town of Amatrice on August 24 (Mw 6.0), and was followed by two subsequent large events near Visso on October 26 (Mw 5.9) and Norcia on October 30 (Mw 6.5), plus a cluster of four events with Mw &amp;gt; 5.0 within few hours on 18 January 2017. The affected area had been monitored before the sequence started by the permanent Italian National Seismic Network (RSNC), and was enhanced during the sequence by temporary stations deployed by the National Institute of Geophysics and Volcanology and the British Geological Survey. By the middle of September, there was a dense network of 155 stations, with a mean separation in the epicentral area of 6–10 km, comparable to the most likely earthquake depth range in the region. This network configuration was kept stable for an entire year, producing 2.5 TB of continuous waveform recordings.Here we describe how this data was used to develop a large and comprehensive earthquake catalogue using the Complete Automatic Seismic Processor (CASP) procedure. This procedure detected more than 450 000 events in the year following the first main shock, and determined their phase arrival times through an advanced picker engine (RSNI-Picker2), producing a set of about 7 million P- and 10 million S-wave arrival times. These were then used to locate the events using a non-linear location (NLL) algorithm, a 1-D velocity model calibrated for the area, and station corrections and then to compute their local magnitudes (ML). The procedure was validated by comparison of the derived data for phase picks and earthquake parameters with a handpicked reference catalogue (hereinafter referred to as ‘RefCat’). The automated procedure takes less than 12 hr on an Intel Core-i7 workstation to analyse the primary waveform data and to detect and locate 3000 events on the most seismically active day of the sequence. This proves the concept that the CASP algorithm can provide effectively real-time data for input into daily operational earthquake forecasts,The results show that there have been significant improvements compared to RefCat obtained in the same period using manual phase picks. The number of detected and located events is higher (from 84 401 to 450 000), the magnitude of completeness is lower (from ML 1.4 to 0.6), and also the number of phase picks is greater with an average number of 72 picked arrival for a ML = 1.4 compared with 30 phases for RefCat using manual phase picking. These propagate into formal uncertainties of ±0.9 km in epicentral location and ±1.5 km in depth for the enhanced catalogue for the vast majority of the events. Together, these provide a significant improvement in the resolution of fine structures such as local planar structures and clusters, in particular the identification of shallow events occurring in parts of the crust previously thought to be inactive. The lower completeness magnitude provides a rich data set for development and testing of analysis techniques of seismic sequences evolution, including real-time, operational monitoring of b-value, time-dependent hazard evaluation and aftershock forecasting.

The Input-Output Equivalent Sources Method for Fast Simulations of Distributed Nonlinearities in Bulk Acoustic Wave Resonators and Filters.

This work presents a new method to analyze weak distributed nonlinear (NL) effects, with a focus on the generation of harmonics (H) and intermodulation products (IMD) in bulk acoustic wave (BAW) resonators and filters composed of them. The method consists of finding equivalent current sources [input-output equivalent sources (IOES)] at the H or IMD frequencies of interest that are applied to the boundary nodes of any layer that can contribute to the nonlinearities according to its local NL constitutive equations. The new methodology is compared with the harmonic balance (HB) analysis, by means of a commercial tool, of a discretized NL Mason model, which is the most used model for NL BAW resonators. While the computation time is drastically reduced, the results are fully identical. For the simulation of a seventh-order filter, the IOES method is around 700 times faster than the HB simulations.

Equilibrium and scattering properties of a nonlocal three-soliton molecule in Bose–Einstein condensates with competing nonlinearities

We carry out numerical and variational investigation of equilibrium and scattering properties of a bright three-soliton molecule in Bose–Einstein condensates with competing local and nonlocal nonlinearities in one-dimensional geometry. Our model predicts that the degree of the nonlocality and the soliton phase may strongly affect the binding energy and the soliton width. We show that the interaction of three solitons depends on their separation distance and on their relative phase. The scattering properties of these composite nonlinear structures by Gaussian potential barrier are analyzed variationally and numerically. It is found that stable transmission and reflection where the molecular structure is preserved can occur only for a specific barrier height and soliton velocity.

Stability and evolution of electromagnetic solitons in relativistic degenerate laser plasmas

The dynamical behaviours of electromagnetic (EM) solitons formed due to nonlinear interaction of linearly polarized intense laser light and relativistic degenerate plasmas are studied. In the slow-motion approximation of relativistic dynamics, the evolution of weakly nonlinear EM envelope is described by the generalized nonlinear Schrödinger (GNLS) equation with local and nonlocal nonlinearities. Using the Vakhitov–Kolokolov criterion, the stability of an EM soliton solution of the GNLS equation is studied. Different stable and unstable regions are demonstrated with the effects of soliton velocity, soliton eigenfrequency, as well as the degeneracy parameter $R=p_{Fe}/m_ec$ , where $p_{Fe}$ is the Fermi momentum and $m_e$ the electron mass and $c$ is the speed of light in vacuum. It is found that the stability region shifts to an unstable one and is significantly reduced as one enters from the regimes of weakly relativistic $(R\ll 1)$ to ultrarelativistic $(R\gg 1)$ degeneracy of electrons. The analytically predicted results are in good agreement with the simulation results of the GNLS equation. It is shown that the standing EM soliton solutions are stable. However, the moving solitons can be stable or unstable depending on the values of soliton velocity, the eigenfrequency or the degeneracy parameter. The latter with strong degeneracy $(R&gt;1)$ can eventually lead to soliton collapse.

Unified Bayesian conditional autoregressive risk measures using the skew exponential power distribution

Conditional Autoregressive Value-at-Risk and Conditional Autoregressive Expectile have become two popular approaches for direct measurement of market risk. Since their introduction several improvements both in the Bayesian and in the classical framework have been proposed to better account for asymmetry and local non-linearity. Here we propose a unified Bayesian Conditional Autoregressive Risk Measures approach by using the Skew Exponential Power distribution. Further, we extend the proposed models using a semiparametric P-Spline approximation answering for a flexible way to consider the presence of non-linearity. To make the statistical inference we adapt the MCMC algorithm proposed in Bernardi et al. (2018) to our case. The effectiveness of the whole approach is demonstrated using real data on daily return of five stock market indices.

Dynamic Analysis of Bolted Cantilever Beam with Finite Element Method Considering Thread Relation

There are complex nonlinear behaviors and mechanisms in the bolted joint interface. Thus, the bolted joint is crucial to the complex nonlinear dynamic response of the structure. However, in the traditional structural dynamic analysis, the screw connection is usually neglected, which makes it challenging to analyze and study the nonlinear dynamic behavior of bolted structures. Hence, based on the Timoshenko beam theory and finite element method, this paper introduces a model considering thread connection to analyze the dynamic response under different excitation. Eventually, the results indicate that owing to the local nonlinearity of bolts, the whole bolted cantilever beam shows hardening-type characteristics. In addition, the frequency response curve also depicts the typical nonlinear phenomenon of instability and uncertainty, namely bifurcation, which preliminarily verifies the correctness and accuracy of the bolted cantilever beam model.

Short-Term Foreshocks as Key Information for Mainshock Timing and Rupture: The Mw6.8 25 October 2018 Zakynthos Earthquake, Hellenic Subduction Zone.

Significant seismicity anomalies preceded the 25 October 2018 mainshock (Mw = 6.8), NW Hellenic Arc: a transient intermediate-term (~2 yrs) swarm and a short-term (last 6 months) cluster with typical time-size-space foreshock patterns: activity increase, b-value drop, foreshocks move towards mainshock epicenter. The anomalies were identified with both a standard earthquake catalogue and a catalogue relocated with the Non-Linear Location (NLLoc) algorithm. Teleseismic P-waveforms inversion showed oblique-slip rupture with strike 10°, dip 24°, length ~70 km, faulting depth ~24 km, velocity 3.2 km/s, duration 18 s, slip 1.8 m within the asperity, seismic moment 2.0 × 1026 dyne*cm. The two largest imminent foreshocks (Mw = 4.1, Mw = 4.8) occurred very close to the mainshock hypocenter. The asperity bounded up-dip by the foreshocks area and at the north by the foreshocks/swarm area. The accelerated foreshocks very likely promoted slip accumulation contributing to unlocking the asperity and breaking with the mainshock. The rupture initially propagated northwards, but after 6 s stopped at the north bound and turned southwards. Most early aftershocks concentrated in the foreshocks/swarm area. This distribution was controlled not only by stress transfer from the mainshock but also by pre-existing stress. In the frame of a program for regular monitoring and near real-time identification of seismicity anomalies, foreshock patterns would be detectable at least three months prior the mainshock, thus demonstrating the significant predictive value of foreshocks.

Families of solitons under nonlocal and PT symmetric competing nonlinearity

In this paper, we discover several new soliton families in the PT symmetric optical lattices with the nonlocal competing cubic–quintic nonlinearity, and investigate their existence and stability ranges. We detailedly study the influence of the degree of nonlocality, the quintic nonlinearity and the PT symmetry on these solitons, and obtain the power surfaces of solitons under different degrees of nonlocality and PT symmetry. We demonstrate that solitons can be linearly stabilized in the first Bloch bandgap under the nonlocal focusing cubic and local defocusing quintic nonlinearity, while they can only exist in the semi-infinite Bloch bandgap under the nonlocal defocusing cubic and local focusing quintic nonlinearity.