Local Nonlinearity Research Articles

An Efficient Reduced-Order Multi-Scale Simulation Approach for Nonlinear Analysis of Frame Structures with Nonlinear Localization

ABSTRACT This paper presents a reduced-order multi-scale simulation approach for nonlinear analysis of structures with local nonlinearities. Using it, the traditional global structural analysis can be changed to that of a reduced-order system where only nodal displacements of nonlinear elements are solved using the Newton–Raphson method in a single iterative procedure. Meanwhile, two nonlinear force/stiffness correctors are introduced based on the stress decomposition to describe material nonlinearities of frame members. In addition, two simulation ways are used to deal with the nonlinearities of the yielding frame members. Finally, the static and transient structural systems are simulated to verify the approach.

An improved model order reduction method for dynamic analysis of large-scale structures with local nonlinearities

In this study, an improved model order reduction method is developed for the dynamic analysis of large-scale structures with local nonlinearities. In this method, the reduced system associated with the elements in nonlinear state is reconstructed adaptively based on the expressions of the element generalized deformation, which is defined by the eigen-decomposition of the element stiffness matrix. With the introduction of a simplified pre-conditioning method, a novel Newton iteration approach with Krylov subspace methods is proposed to solve the reduced system efficiently. Thus, dynamic analysis can be conducted efficiently for the entire structure. To expand the scope of application, an alternating iterative solution algorithm based on the model order reduction method is further developed for the structural systems simulated with both pseudo-static models and parameterized dynamic models. According to the results of numerical investigation, the novel Newton iteration approach significantly improves the solution efficiency of dynamic analysis for the large-scale structures with local nonlinearities. Meanwhile, the implementation of the alternating iterative solution algorithm contributes a new idea to the fast solution of structural systems with various types of nonlinear models.

IBEM-FEM coupling method for full process nonlinear ground motion simulation of near-fault sedimentary basins

Based on the concept of domain reduction, a two-step method combining the indirect boundary element method (IBEM) and the finite element method (FEM) was developed to simulate the full-process ground motion of a 3D (three-dimensional) near-fault complex site. In the first step, the seismic response of dislocation motion in semi-infinite space crust model was accurately solved using the IBEM, which can significantly reduce the dimensionality of the numerical model and easily achieve a computational frequency of 5 Hz on an ordinary workstation. In the second step, considering the inhomogeneity and nonlinearity of soft soils, the seismic wave scattering from a 3D local site was finely simulated using a FEM. The external seismic response obtained through IBEM is converted into equivalent load and transferred to the internal finite element domain, while the PML boundary is applied to absorb the pseudo reflected wave on the truncated boundary. Then, the reliability and accuracy of the method was demonstrated by comparing with the results of existing studies and the traditional IBEM. Further, the proposed coupled method was used to investigate the seismic response of a large near-faulted sedimentary basin. The numerical investigation focuses on four aspects: acceleration histories and response spectra, wave-field snapshot, duration of ground motion, and peak magnitudes and response spectrum of velocity. The results highlight the influence of local site soil non-linearity and medium impedance ratios on the analysis of near-fault seismic response. This research is essential for seismic zoning of complex sedimentary basin sites near faults, and also establishes a “bridge” for seismic design from faults to structures.

The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation

We consider the non-semisimple 0:1 resonance (i.e., the unperturbed equilibrium has two purely imaginary eigenvalues ( and ) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), this bifurcation essentially triggered by the reversible universal unfolding with respect to BCKV-restricted morphisms of the planar non-semisimple singularity (the is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the -symmetry generated by the additional pair of imaginary eigenvalues to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g., 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by Broer et al. (Z Angew Math Phys 44: 389-432, 1993). An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.

Data-driven structural identification of nonlinear assemblies: Structures with bolted joints

The identification of nonlinearities that have a significant impact on dynamic behaviour of complex mechanical structures is necessary for ensuring structural efficiency and safety. A new methodology for structural identification of nonlinear assemblies is proposed in this paper that enables the discovery of stiffness and damping nonlinear models especially when it is not possible to directly measure the degrees of freedom where non-trivial nonlinearities are located. Input-output time-domain data collected at accessible locations on the structure are used to learn nonlinear models in the unmeasured locations. This is accomplished by making use of virtual sensing and model reduction schemes along with a physics-informed identification method recently developed by the authors (Safari and Londoño 2021). The methodology is suited for weakly nonlinear systems with localised nonlinearities for which their location is assumed to be known. It also takes into account dominant modal couplings within the identification process. The proposed methodology is demonstrated on a case study of a nonlinear structure with a frictional bolted joint, in numerical and experimental settings. It is shown that the model selection and parameter estimation for weakly nonlinear elements can be carried out successfully based on a reduced-order model which includes only a modal equation along with relevant modal contributions. Using the identified localised nonlinear models, both the reduced and full-order models can be updated to simulate the dynamical responses of the structure. Results suggests that the identified nonlinear model, albeit simple, generalises well in terms of being able to estimate the structural responses around modes which were not used during the identification process. The identified model is also interpretable in the sense that it is physically meaningful since the model is discovered from a predefined library featuring different nonlinear characteristics.

Two-Dimensional Gap Solitons in Parity-Time Symmetry Moiré Optical Lattices with Rydberg–Rydberg Interaction

Realizing single light solitons that are stable in high dimensions is a long-standing goal in research of nonlinear optical physics. Here, we address a scheme to generate stable two-dimensional solitons in a cold Rydberg atomic system with a parity-time (PT) symmetric moiré optical lattice. We uncover the formation, properties, and their dynamics of fundamental and two-pole gap solitons as well as vortical ones. The PT symmetry, lattice strength, and the degrees of local and nonlocal nonlinearity are tunable and can be used to control solitons. The stability regions of these solitons are evaluated in two numerical ways: linear-stability analysis and time evolutions with perturbations. Our results provide an insightful understanding of solitons physics in combined versatile platforms of PT-symmetric systems and Rydberg–Rydberg interaction in cold gases.

Seismic mitigation of a benchmark twenty-story steel structure based on intermodal targeted energy transfer (IMTET)

This study investigates a new intermodal targeted energy transfer (IMTET) concept for rapid, effective, and purely passive seismic mitigation of a benchmark large-scale model of a twenty-story steel structure. IMTET is based on extremely rapid nonlinear scattering of seismic input energy from low to high frequency modes of a building. This effect is achieved by introducing strategically placed, local strong vibro-impact nonlinearities that are generated by contacts of the building floors with a relatively light, yet stiff, auxiliary core structure. Accordingly, the performance of IMTET is studied here, with the benchmark structure realized through a set of previously established performance criteria. The seismic loads are simulated based on records from three historical earthquakes, namely the 1995 Kobe, 1994 Northridge, and 1940 El Centro. To assess the robustness of the proposed passive nonlinear mitigation mechanism, the clearance distributions as well as the core structure parameters, are optimized for a specific seismic excitation (Kobe). Subsequently, the optimized design is tested against the two other historical earthquake records to demonstrate the effectiveness of the IMTET for these cases as well. The numerical results show that the vibro-impacts rapidly, robustly, and almost irreversibly redistribute the seismic input energy from low to high frequency structural modes, thus realizing a highly effective passive earthquake protective system. In addition, when optimized, this new concept can be realized fully passively, without the need to add any mass to the building, and at the cost of only moderate increases in the resulting floor accelerations and local stresses. In addition, the nonlinear vibro-impacts between the floors and the core structure reduce the seismic input energy to the building compared to the no core case, adding an additional benefit to the seismic mitigation approach. Therefore, the IMTET methodology for seismic mitigation has the potential to significantly enhance the seismic performance of building structures.

Finite element modelling of bridge piers subjected to eccentric load combined with reinforcement corrosion

Corrosion of reinforcing steel in reinforced concrete infrastructure has been identified as the leading cause of concrete deterioration in North America. As the most critical bridge element, piers are more susceptible to reinforcement corrosion, especially in coastal areas and cold regions due to the application of de-icing salts. The combination of second-order effects and local nonlinearity dictates the failure mode of slender and eccentrically loaded columns. Degradation effects would be more critical in the presence of corrosion, at which the nonlinearity effects become more dominant. A comprehensive corrosion damage model is adopted from the literature in the present study. A stage-based corrosion scenario is then proposed to study the detrimental corrosion effects on the ultimate capacity, rigidity, and ductility of bridge piers subjected to eccentric load through finite element analysis. The interaction diagram of the corroded pier is then generated at different corrosion stages to introduce new safety margins. The accuracy of the proposed model is investigated by comparisons with available test data from the literature.

A non-intrusive dynamic sensitivity-based substructure model updating method for nonlinear systems

In nonlinear finite element model updating (FEMU), one of the main challenges is how to improve the updating efficiency for large-scale structures. The computational time of updating depends on the degree-of-freedom of the finite element model, and the optimization algorithm that the user chooses. In this work, a novel dynamic response sensitivity-based nonlinear FEMU method using the substructures is presented to reduce the time consuming of computation and updating. Considering the residual structural modes, the enhanced Hurty/Craig-Bampton (EHCB) substructure method is developed to establish a more precise initial numerical model, which also has high computational efficiency. A non-intrusive algorithm based on the real and imaginary perturbation of updating parameters is implemented to compute the nonlinear substructural response sensitivities, and further reduce the iteration loops in the nonlinear FEMU process. Simulation examples with smooth and non-smooth localized nonlinearities are conducted to verify the accuracy, robustness, and efficiency of the proposed method. Results indicated that the proposed method can effectively and efficiently update the nonlinear finite element model with high accuracy.

Spatial Self-Phase Modulation in Graphene-Oxide Monolayer

The spatial self-phase modulation (SSPM) of the optical field revealed the magnitude and polarity of nonlinear refraction coefficients of the graphene-oxide (GO) atomic layers in an aqueous base solution with a resonant excitation using a chopped quasi-static laser at 532 nm. The SSPM of the optical field as a result of the intrinsic nonlinear refraction coefficient of GO atomic layers and the spatial distribution of intensity displayed the concentric diffraction rings at the far field due to the coherent superposition of transverse wave vectors. The number of concentric rings as a function of the applied intensity revealed the nonlinear refraction coefficient of GO which was estimated to be ~–6.65 × 10−12 m2/W for the laser-excitation duration of ~0.32 s, where the negative polarity of nonlinear refraction coefficient was confirmed with the interference image profile of SSPM. The upper and vertical distortion of concentric rings at the far field at the longer laser-excitation duration of ~0.8 s indicates the distortion of the coherent superposition of transverse wave vectors due to the localized thermal vortex of GO in the aqueous solution that offers novel platforms of thermal metrology based on localized optical nonlinearity and temperature-sensitive all-optical switching.

Concurrent multiscale analysis of anti-seepage structures in embankment dam based on the nonlinear Arlequin method

The scales of embankment dam and local anti-seepage structures (core wall or cut-off wall) differ heavily, which poses challenges to analyze both the global and local responses simultaneously. In this study, the nonlinear Arlequin method is introduced to evaluate the behavior of anti-seepage structures in embankment dam within a concurrent multiscale framework. Based on the energy blending approach, the governing equations are deduced, in which the material nonlinearity effect and the Arlequin coupling effect can be separated from each other. Then, a list of subroutines of ABAQUS is used to implement the proposed framework jointly. Multiscale analysis of elasto-plastic cantilever beam demonstrates the effectiveness of the proposed method. The effects of coupling operators are discussed, and constant weighting functions are recommended for nonlinear coupling analysis. The analyses of embankment dams indicate that the uneven settlement pattern and the resulted arching effect between the core and the shell as well as the complicated bending deformation state of cut-off wall can be well captured by the proposed method. It is affirmed that the nonlinear Arlequin method facilitates the increase of computational efficiency without losing much accuracy, showing great merits of modeling local nonlinearity and global response simultaneously under a concurrent multiscale framework.

Machine Learning Non-Reciprocity of a Linear Waveguide With a Local Nonlinear, Asymmetric Gate: Case of Strong Coupling

Abstract We study nonreciprocity in a passive linear waveguide augmented with a local asymmetric, dissipative, and strongly nonlinear gate. Strong coupling between the constituent oscillators of the waveguide is assumed, resulting in broadband capacity for wave transmission. The local nonlinearity and asymmetry at the gate can yield strong global nonreciprocal acoustics, in the sense of drastically different acoustical responses depending on which side of the waveguide a harmonic excitation is applied. Two types of highly nonreciprocal responses are observed: (i) Monochromatic responses without frequency distortion compared to the applied harmonic excitation, and (ii) strongly modulated responses (SMRs) with strong frequency distortion. The complexification averaging (CX-A) method is applied to analytically predict the monochromatic solutions of this strongly nonlinear problem, and a stability analysis is performed to study the governing bifurcations. In addition, we build a machine learning framework where neural net (NN) simulators are trained to predict the performance measures of the gated waveguide in terms of certain transmissibility and nonreciprocity measures. The NN drastically reduces the required simulation time, enabling the determination of parameter ranges for desired performance in a high-dimensional parameter space. In the predicted desirable parameter space for nonreciprocity, the maximum transmissibility reaches 40%, and the transmitted energy varies by up to three orders of magnitude depending on the direction of wave transmission. The machine learning tools along with the analytical methods of this work can inform predictive designs of practical nonreciprocal waveguides and acoustic metamaterials that incorporate local nonlinear gates.

Can transient simulation efficiently reproduce well known nonlinear effects of jointed structures?

Although joints are omnipresent in engineering structures, their proper consideration in simulations is still a challenging area of research. This is because joints introduce local nonlinearities which lead to load and preload-dependent resonance frequencies and damping, mode coupling, high variability of contact area and contact pressures. The knowledge about those phenomena has been gathered via systematic experimental testing. A meaningful digital or virtual replication of such tests would require a transient numerical time integration, where besides all dynamic effects, the local and fine discretized nonlinear contact and friction forces in the joint also need to be considered. The Finite Element Method can only provide this theoretically since the necessary computation times make practical use almost impossible. Newer methods, where the joint deformation is represented via so-called contact modes instead of nodal degrees of freedom, offer a perspective for transient virtual tribomechadynamics. This paper is devoted to the following question: Is it possible for the first time to numerically reproduce real experiments with low computing times so that the previously mentioned nonlinear effects can be observed? In this work the transient numerical response of a jointed structure due to different loads (sine sweep, sine, step and impulse) is evaluated and related to measurements. The application of the Hilbert Transform to the time data, as known from measurements, reveals load and preload dependent eigenfrequency and damping. Other measured phenomena like mode coupling, a high variability in the contact area and resulting contact force can also be observed numerically. The results are in good qualitative agreement with the experiments. This work proves the principal capability of virtual tribomechadynamics as an efficient complement to real testing for gaining deeper insight. All the powerful measurement methods can directly be applied to simulated data.

A high-order finite element continuation for buckling analysis of porous FGM plates

This work presents an effective approach to investigating the buckling and post-buckling behavior of porous FGM plates. The main objective of the present analysis is to use a High Order Continuation based on the Asymptotic Numerical Method with the Finite Element Method for nonlinear behaviors of a Porous Functionally Graded Material plate (PFGM) with different porosity distributions under various types of transverse loads. In our study, we will take into account the lack of uniform and non-uniform porosity using a Higher-order Shear Deformation Theory (HSDT). The adapted solver proves to be more effective for analyzing non-linear mechanical phenomena and more precisely the post-buckling analysis of the PFGM plates. Indeed, this numerical high order algorithm presents an adaptive step length to the local nonlinearity of the solution branch and to detect easily the bifurcation points and bifurcated branches. When compared to iterative techniques like Newton–Raphson, the proposed high-order algorithm has the benefit of speeding up the resolution of nonlinear problems. The mathematical formulation is based on the higher-order shear deformation plate theory taking into consideration geometrical nonlinearity and initial imperfections. Numerical results of a parametric study on the effect of various types of load, porosity distribution, grading index, porosity index and different span-to-thickness ratios will be investigated.

Two-dimensional spatial optical solitons in Rydberg cold atomic system under the action of optical lattice

Realizing stable high-dimensional light solitons is a long-standing goal in the study of nonlinear optical physics. However, in high-dimensional space, the light field will inevitably be distorted due to diffraction. In order to solve the diffraction effect in nonlinear Kerr media and achieve the spatial localization of light fields, we propose a scheme to generate stable two-dimensional (2D) solitons in a cold Rydberg atomic system with a Bessel optical lattice, where a three-level atomic structure, a weak probe laser field, and a strong control field constitute the Rydberg-dressed atomic system. When the local nonlinearity, Bessel potential, and nonlocal nonlinearity which is caused by the long-range Rydberg-Rydberg interaction (RRI) between Rydberg atoms are balanced, the probe field can be localized. Under the approximation of electric dipole and rotating wave, the stable solution of probe field is obtained by solving Maxwell-Bloch equations numerically. A cluster of 2D spatial solitons, including fundamental, two-pole, quadrupole and vortex solitons, is found in this system. Among them, the fundamental, dipole and quadrupole have, one, two, and four intensity centers, respectively. Vortex solitons, on the other hand, exhibit vertical characters in profiles and phase structures. The formation and transmission of these solitons can be controlled by system parameters, such as the propagation coefficient, the degree of nonlocal nonlinearity, and Bessel lattice strength. The stable regions of these solitons are determined by anti Vakhitov Kolokolov (anti-VK) criterion and linear stability analysis method. It is found that four kinds of solitons can be generated and stably propagate in space with proper parameters. Owing to the different structures of the poles, the fundamental state and vortex state remain stable, while the quadrupole ones are unstable. In the modulation of solitons, there is a cutoff value of propagation constant <inline-formula><tex-math id="M1">\begin{document}${b_{{\text{co}}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.png"/></alternatives></inline-formula>, only below which value, the solitons can propagate stably. The light intensity of soliton shows a periodic behavior by tuning Bessel lattice strength. The period of the intensity decreases with the order of the solitons as a result of the interaction between the poles. It is also found that the solitons are more stable with weak nonlocal nonlinearity coefficient. This study provides a new idea for the generation and regulation of optical solitons in high dimensional space.

Variational approach for an elastic beam equation with local nonlinearities

We discuss the existence of at least one, two and three weak solutions for a class of elastic beam differential equations with two parameters $\lambda $ and $\mu $, using variational methods and critical point theory.

Structurally determinable optical soliton molecules enabled by a pre-designed microfiber resonator in a passively mode-locked fiber laser

The temporal structure of an optical soliton molecule (SM) produced in a passively mode-locked fiber laser relies directly on phase relations among the comprised solitons. However, such phase relations are typically determined by many global and local parameters of the fiber cavity. Thus, any parameter disturbance would exert complex distortion on the SM. This is always unpredictable and difficult to manipulate. One factor is the difficulty in deciding what parameter options have enabled the SM’s formation. The second is the coupling effect among the related parameters. Consequently, the produced SMs usually exhibit considerable uncertainty and poor stability. How to obtain pre-defined SMs has long been a sought-after yet technically unsolved issue. Herein, as a preliminary investigation we demonstrate that employing a microfiber knot resonator (MKR) in a mode-locked fiber laser can enable the formation of artificially defined and structure-stabilized SMs. Specifically, the MKR enables the single soliton splitting into several ones through enhancing the local nonlinearity. But more importantly, it meanwhile functions as a notch filter that dominates and tailors the spectral evolution. The tailored spectrum is then mapped to the temporal domain, grouping the randomly split solitons into a structured and stabilized SM. Our results suggest an easy-to-access avenue in producing structure-determinable SMs in fiber lasers.

Sign-Changing Solutions for Schrödinger-Poisson System with Local Nonlinearity

Sign-Changing Solutions for Schrödinger-Poisson System with Local Nonlinearity

Nonlinear vibration analysis and stability analysis of rotor systems with multiple localized nonlinearities

This study proposes a methodology to analyze the nonlinear vibration characteristics of rotor systems with multiple localized nonlinearities adopting the Finite Element Method (FEM), free interface Component Mode Synthesis (CMS) method, and modified Incremental Harmonic Balance (IHB) method. The rotor system is supported by squeeze film dampers (SFDs) on both sides, and at the nodes of the SFD arrangement, strong local nonlinearities will appear due to fluid-film forces. The methodology to analyze the nonlinear vibration characteristics of the system by reducing the degree of freedom of the rotating system with multiple local nonlinear factors and combining with the IHB method is proposed for the first time in this paper. The FEM is used to write motion equations in components, and the CMS method is applied to reduce the degrees of freedom of linear components. The IHB method is used to solve the motion equations of the nonlinear system. The system has one linear component and two nonlinear components. For linear components, modal coordinates are used, and for nonlinear components, the original physical coordinate system is used. By synthesizing these three components, the motion equation of the whole system is created. In order to validate the effectiveness of the method, the results obtained by the proposed method are compared with the data in the published literature, and the system responses are considered when specific parameters are changed. The stability analysis of the calculated solutions is carried out using the Floquet theory.

Transient Response Sensitivity Analysis of Localized Nonlinear Structure Using Direct Differentiation Method

Based on the direct differentiation method, sensitivity analysis of transient responses with respect to local nonlinearity is developed in this paper. Solutions of nonlinear equations and time-domain integration are combined to compute the response sensitivities, which consist of three steps: firstly, the nonlinear differential equations of motion are solved using Newton–Raphson iteration to obtain the transient response; secondly, the algebraic equations of the sensitivity are obtained by differentiating the incremental equation of motion with respect to nonlinear coefficients; thirdly, the nonlinear transient response sensitivities are determined using the Newmark-β integration in the interested time range. Three validation studies, including a Duffing oscillator, a nonlinear multiple-degrees-of-freedom (MDOF) system, and a cantilever beam with local nonlinearity, are adopted to illustrate the application of the proposed method. The comparisons among the finite difference method (FDM), the Poincaré method (PCM), the Lindstedt–Poincaré method (LPM), and the proposed method are conducted. The key factors, such as the parameter perturbation step size, the secular term, and the time step, are discussed to verify the accuracy and efficiency. Results show that parameter perturbation selection in the FDM sensitivity analysis is related to the nonlinear features depending on the initial condition; the consistency of the transient response sensitivity can be improved based on the accurate nonlinear response when a small time step is adopted in the proposed method.