Nonlinear Source Research Articles

Equivalent Near-Field Corner Frequency Analysis of 3D Dynamic Rupture Simulations Reveals Dynamic Source Effects

Abstract Dynamic rupture simulations generate synthetic waveforms that account for nonlinear source and path complexity. Here, we analyze millions of spatially dense waveforms from 3D dynamic rupture simulations in a novel way to illuminate the spectral fingerprints of earthquake physics. We define a Brune-type equivalent near-field corner frequency (fc) to analyze the spatial variability of ground-motion spectra and unravel their link to source complexity. We first investigate a simple 3D strike-slip setup, including an asperity and a barrier, and illustrate basic relations between source properties and fc variations. Next, we analyze >13,000,000 synthetic near-field strong-motion waveforms generated in three high-resolution dynamic rupture simulations of real earthquakes, the 2019 Mw 7.1 Ridgecrest mainshock, the Mw 6.4 Searles Valley foreshock, and the 1992 Mw 7.3 Landers earthquake. All scenarios consider 3D fault geometries, topography, off-fault plasticity, viscoelastic attenuation, and 3D velocity structure and resolve frequencies up to 1–2 Hz. Our analysis reveals pronounced and localized patterns of elevated fc, specifically in the vertical components. We validate such fc variability with observed near-fault spectra. Using isochrone analysis, we identify the complex dynamic mechanisms that explain rays of elevated fc and cause unexpectedly impulsive, localized, vertical ground motions. Although the high vertical frequencies are also associated with path effects, rupture directivity, and coalescence of multiple rupture fronts, we show that they are dominantly caused by rake-rotated surface-breaking rupture fronts that decelerate due to fault heterogeneities or geometric complexity. Our findings highlight the potential of spatially dense ground-motion observations to further our understanding of earthquake physics directly from near-field data. Observed near-field fc variability may inform on directivity, surface rupture, and slip segmentation. Physics-based models can identify “what to look for,” for example, in the potentially vast amount of near-field large array or distributed acoustic sensing data.

Exploration of nonlinear radiative heat energy on Buongiorno modeled nano liquid toward an inclined porous plate with heat source and variable chemical reaction

Nanofluids are fluid suspensions of nanoparticles that exhibit a considerable improvement in their characteristics at low nanoparticle concentrations. Numerous research on nanofluids focuses on interpreting their behavior in order to use them in applications where improving straight heat transmission is crucial, such as in various industrial settings, nuclear reactors, transportation, biology, food, and electronics. Thus, this study examines the implementation of a novel numerical technique, namely the shooting method for nonlinear radiative heat energy study on Buongiorno modeled nano liquid confined by an inclined porous plate. The Brownian and thermophoresis diffusions impacts are also accounted. The transmission of thermal and solutal energy is regulated by the considerable influence of nonlinear thermal radiation, heat source, and variable chemical reactions. The dimensional modeled partial differential equations (PDEs), by using precise similarity functions, have been mutated into ordinary differential equations (ODEs). The outcomes for the flow field, thermal, and solutal outlines are captured graphically. The values of the dimensionless parameters are chosen from the literature in such a way that they have significantly affected the dimensionless boundary layer (BL) profiles. Also, the non-dimensional profiles of velocity, temperature, and concentration observe two different trends (increasing and decreasing) for diverse values of the dimensionless parameters.

On nonlinear Sobolev equation with the Caputo fractional operator and exponential nonlinearity

The initial value problem for the Caputo type time‐fractional Sobolev equation with a nonlinear exponential source function is investigated in this work. We establish the existence and uniqueness of mild solutions corresponding to two different initial data assumptions. We derive global results of a unique mild solution with small initial data using some Sobolev/Sobolev‐Orlicz embeddings, a weighted Banach space, and the fixed point theorem. In the absence of any smallness assumptions, the Cauchy iteration method demonstrates that the mild solution blows up at a finite time or exists globally in time. Finally, we consider some illustrated examples to test the results obtained in theory.

Generation of Reflected Second Harmonic at Semiconductor Quantum Dots

Generation of the reflected second harmonic has been observed for the first time in a composite material: a glass matrix containing microscopic CdSe crystals, ranging in size from 500 to 50 Å. The violation of polarization selection rules in the second-harmonic generation indicates that the nonlinear sources are of a fluctuational nature. A size effect has also been observed: The intensity of the second harmonic increases by a factor of 102 as the size of the microscopic crystals is reduced. A possible mechanism for this size effect is analyzed.

AI Models for Supporting SI Analysis on PCB Net Structures: Comparing Linear and Non-Linear Data Sources

Abstract. Signal integrity (SI) is an essential part in assuring the functionality of microelectronic components on a printed circuit board (PCB). Depending on the complexity of the designed interconnect structure, even the experienced PCB developer might be reliant on multiple design cycles to optimally configure the PCB parameters, which eventually results in a very complex, time-consuming and costly process. Under these aggravating conditions, artificial intelligence (AI) models may have the potential to support and simplify the SI-aware PCB design process by building predictive models and proposing design solutions to streamline the existing workflows and unburden the PCB designer. In this paper, the AI approach is divided into two separate stages consisting of neural network (NN) regression in the first step and parameterization of the PCB net structure in the second step. First, the NN models are applied to learn the relationship between the electrical parameters and the resulting signal quality captured by domain-oriented signal features in the time domain. Second, based on the trained NN models, on the one hand, the k-nearest neighbor (kNN) method is utilized to select solution candidates within the feature space, while on the other hand, genetic algorithms (GA) are applied to directly optimize the parameters of the interconnect structure. Moreover, the influence of the simulation abstraction level is investigated by comparing simulation data originating from linear and I/O buffer information specification (IBIS)-based non-linear modeling of the integrated circuit (IC) characteristics concerning the prediction accuracy and direct transferability. Finally, transfer learning concepts are evaluated to exchange learned knowledge representations between the different modeling of the IC characteristics to improve data efficiency and reduce computational complexity.

Reaction-Diffusion Systems with Nonlinear Sources of Different Intensities in the Case of Multiple Root without Quasimonotonicity Condition

Reaction-Diffusion Systems with Nonlinear Sources of Different Intensities in the Case of Multiple Root without Quasimonotonicity Condition

Study of a Transmission Problem with Friction Law and Increasing Continuous Terms in a Thin Layer

The aim of this paper is to establish the asymptotic analysis of nonlinear boundary value problems. The non-stationary motion is given by the elastic constructive law. The contact is described with a version of Tresca’s law of friction. A variational formulation of the model, in the form of a coupled system for the displacements and the nonlinear source terms, is derived. The existence of a unique weak solution of the model is established. We also give the problem in transpose form, and we demonstrate different estimates of the displacement and of the source term independently of the small parameter. The main corresponding convergence results are stated in the different theorems of the last section.

Global dynamics in a chemotaxis system involving nonlinear indirect signal secretion and logistic source

Global dynamics in a chemotaxis system involving nonlinear indirect signal secretion and logistic source

Blow-Up of Solution of Lamé Wave Equation with Fractional Damping and Logarithmic Nonlinearity Source Terms

In this work, by the use of a semigroup theory approach, we provide a global solution for an initial boundary value problem of the wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. In addition to this, we establish a blow-up result for the solution under the condition of non-positive initial energy.

Numerical Solution of Hydrodynamic Efficiency Equations for an Oscillating Water Column Wave Energy Converter Using the Method of Fundamental Solutions

The oscillating water column (OWC) converter, a prominent and widely adopted device for capturing wave energy, has yet to attain commercialization due to its intricate hydrodynamic behavior. Consequently, numerous laboratory and numerical studies have been carried out on OWC converters. One of the significant parameters in the study and comparison of different converters is their hydrodynamic efficiency parameter. Hence, the primary objective of this study is to examine the efficiency of an oscillating water column converter using the meshless method of fundamental solutions (MFS) to solve the governing equations. To address the challenges posed by the low wave steepness and to optimize computational efficiency, a complete linearization of two nonlinear sources at the boundary conditions inside and outside the OWC chamber was undertaken. Next, the boundary conditions were temporally discretized using three distinct methods: the Newmark, the forward differentiation formula, and the Runge-Kutta method of the fourth order. Subsequently, a comparison was made among these methods, revealing that the Newmark method exhibited a higher convergence rate to the solution. Using the developed numerical model, a more precise examination of the converter's efficiency, under various sloped bottom geometries, was conducted, surpassing the level of accuracy achieved in prior studies. Two key parameters, namely the opening, and slope, were identified as critical factors influencing the efficiency of converters considering sloped bottoms. The results indicated that the efficiency of the converter decreases with an increase in the slope and a decrease in the opening. Furthermore, to effectively interpret the efficiency of the converter, a suitable parameter was introduced the disparity in the free surface elevation inside and outside the chamber. This parameter proved valuable in providing insights into the converter's performance.

WAVEx: Stabilized finite elements for spectral wind wave models using FEniCSx

The prediction of the wind wave spectrum of the ocean using numerical models are an important tool for researchers, engineers, and communities living in coastal areas. The governing equation of the wind wave models, the Wave Action Balance Equation, presents unique challenges for implementing reliable numerical models because it is highly advective, highly nonlinear and high dimensional. Historically, most operational models have utilized finite difference methods, others have used finite volume methods but relatively few attempts at using finite element methods. In this work, we seek to fill this gap by investigating several different finite element discretizations of the Wave Action Balance Equation. The methods, which include streamline upwind Petrov–Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented and convergence properties are examined for some simplified 2-D test cases. Then, a new spectral wind wave model, WAVEx, is formulated and implemented for the full problem setting. WAVEx uses continuous finite elements along with SUPG stabilization in geographic/spectral space that allows for fully unstructured triangular meshes in both geographic and spectral space. For propagation in time, a second order fully implicit finite difference method is used. When source terms are active, a second order operator splitting scheme is used to linearize the problem. In the splitting scheme, propagation is solved using the implicit method and the nonlinear source terms are treated explicitly. Several test cases, including analytic tests and laboratory experiments, are demonstrated and results are compared to analytic solutions, observations, as well as output from another model that is used operationally.

Existence and nonexistence of solution of fractional Lamé wave equation with polynomial nonlinearity source terms

In this work, we examine the existence and nonexistence of solution for lamé wave equation with a nonlinear source terms and fractional boundary dissipation. By using the semi group theory approach, we prove the existence of a global solution. Furthermore, we established the blow-up results under the non-positive initial energy condition.

Characterization of multiple debris cloud impacts-induced pitting damage evolution in spacecraft structures using temperature-dependent ultrasonic nonlinearity

Facing the harsh cryogenic-elevated temperature condition, evaluation of the debris cloud-induced pervasive but highly complex pitting damage in manned spacecraft structures, featuring multitudinous small-scale craters and cracks disorderedly scattered over a wide region, accompanied by a diversity of micro-damages, is hitherto still a challenging task. In line with the fact that the use of ultrasonic nonlinearity is restricted by its intrinsic vulnerability to measurement contamination, in particular temperature fluctuations. With this motivation, an insight into the modulation mechanism of typical elastic parameters of material at macroscopic under varying temperatures, as well as microscopic interatomic potential, is achieved via theoretical analysis. Based on this, temperature sensitivity coefficients (TSCs) of these elastic parameters are obtained, and a nonlinear temperature sensitivity index (TSI) is established. On this basis, an active TSI-based evaluation method is proposed to improve the accuracy and reliability of pitting damage evaluation by controlling detection temperature (i.e., exploitation of the benign aspects of the temperature sensitivity of ultrasonic nonlinearity), rather than eliminating or compensating for its adverse effect, when other nonlinear sources interference. This method is experimentally corroborated, in which the multiple debris cloud impacts-induced accumulated pitting damage are accurately and intuitively characterized through temperature control, in terms of its severity and distribution characteristics, which is consistent with the actual results. These findings provide an active structural health monitoring (SHM) strategy for promoting the practical application of this proposed approach for characterizing pitting damage evolution in manned spacecraft in qualitative and semi-quantitative manners.

On the quality of stability and bifurcation sets in rotors with permanent shaft bow on nonlinear supports

The paper investigates the dynamics of simple and complex geometry rotors of various configurations with permanent bow, mounted on nonlinear bearings, in regards to nonlinear stability, quality of motion, and bifurcation set. Permanent shaft bow is a fault existing in most slender rotors in thermal machines, e.g. gas turbines, steam turbines, and turbopumps, and other applications of rotating machinery like transmission shafts and geared shafts. In this work, finite element models of multi-segment rotors are implemented, including permanent bow, and a generic rotor system of a uniform rotor with three masses is studied first on its stability characteristics for various bow values, and for different slenderness. A realistic rotor of complex geometry is used as the second application. The rotors are mounted on two plain short bearings to include the nonlinearity of impedance forces with analytical formulas; this is the unique source of nonlinearity in the systems.The rotor-bearing systems are found to produce the well-known oil whirl instability through a Hopf, or a secondary Hopf bifurcation, depending the unbalance grade, in speeds higher than the first critical speed, when bow is not included. Supercritical and subcritical Neimark-Sacker and period doubling bifurcations are produced in speeds lower than critical speed when bow is included above some threshold values. Stable and unstable limit cycles are generated both for unbalanced or balanced rotors with bow, and their periodicity is investigated. A bifurcation set of the rotor-bearing systems is evaluated utilizing the pseudo-arc length continuation method with embedded collocation method, for the evaluation of periodic limit cycles (where these appear). Neimark-Sacker bifurcations are found to collide and vanish as the bow magnitude changes from high to low values. Bow phase angle with respect to unbalance force is found not to influence the respective bifurcation sets.

Existence of non-negative periodic solutions for a degenerate anisotropic parabolic equation with strongly nonlinear source

Abstract The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations.

Modulation instability of incoherent beams revisited.

We examine the spatial modulation instability (MI) of a partially incoherent laser beam. We show that the P < (a/rc)2P0 criterion of beam stability, with a laser peak power P, beam radius a, correlation radius rc, and critical power of self-focusing P0, is applicable only to a limited class of MIs, viz., MIs that can be described as instabilities of a pertinent transverse correlation function found as a solution to the evolution equation, where the expectation of the four-field-product nonlinear source term is factorized as a product of the field intensity and a two-point transverse correlation function. When extended to a more general class of MIs, field evolution analysis of partially coherent beams suggests that MIs can be attenuated, but never completely suppressed. We show that spatial incoherence can lower the MI-buildup rate, thus helping avoid MI-induced beam breakup in physical settings where the MI-buildup length lMI can be kept longer than the length of the nonlinear medium L. Because the lMI > L condition sets a limitation on the field intensity rather than the laser peak power, MI-induced beam breakup can be avoided, even at laser peak powers well above the critical power of self-focusing P0.

The acoustic signature of a rudder in the wake of a propeller: Comparison between infinite and semi-infinite approximations

The acoustic analogy is adopted to reconstruct the sound pressure levels from a propeller–rudder system, using the database of instantaneous realizations of the solution from Large-Eddy Simulations, conducted on a cylindrical grid of 3.8 billion points. Results are compared across incidence angles and rudder geometries, approximated by hydrofoils of infinite and semi-infinite spanwise extents. The latter parameter does not affect significantly the acoustic pressure at no incidence, but its influence grows for increasing angles. Actually, trends are different at inner and outer radii. In the vicinity of the propeller wake, the acoustic signature is dominated by the non-linear sources of sound and is higher when the infinite approximation of the rudder geometry is considered, due to the massive separation affecting the suction side of the hydrofoil. In contrast, moving away along the radial direction the lead goes to the linear sources of sound on the surface of the hydrofoil, which are higher in the case of the semi-infinite approximation of the rudder. Surprisingly, separation phenomena, more significant on the suction side of the infinite hydrofoil, result in a lower loading component of the sound, at least in the range of low frequencies considered in the present study, maybe due to a shift of the acoustic signature of the system towards higher frequencies.

Variable thermal conductivity and chemical reaction aspects in MHD tangent hyperbolic nanofluid flow over an exponentially stretching surface

This work objective focuses on studying the combined influences of variable thermal conductivity, chemical reaction, and magnetohydrodynamics (MHD) on the flow of a tangent hyperbolic nanofluid flow over an exponentially stretching surface, considering a first-order velocity slip condition. Additionally, thermophoresis and Brownian motion impacts are taken into account. The phenomena of heat transfer are analyzed considering several factors such as thermal radiation, Joule heating and nonlinear heat source. On the other hand, mass transfer is explored under the effect of chemical reaction. Tangent hyperbolic fluid is an important branch of non-Newtonian fluids known for its ability to describe shear thinning effects. Understanding fluid flow on exponentially stretched surfaces is of great significance due to its applications in various industrial processes. These applications include fluid film condensing methods, plastic production for making plastic covers, fiber manufacturing (where it is used to spin fibers), glass blowing, metallurgical procedures, and the paper industry. The concept of magnetohydrodynamics (MHD) is significant due to its various engineering applications, such as MHD generators, flow meters, heat reservoirs, small components in different devices, and cooling systems for nuclear reactors. To analyze the system, using similarity transformations, the governing equations of continuity, velocity, and concentration are transformed into non-dimensional differential equations. The numerical solution is obtained using the shooting technique. The study presents the physical significance of all the fluid parameters involved, focusing on the velocity, temperature, and concentration profiles. These profiles are presented graphically and discussed in detail. The results show that the fluid velocity profile increases with enhancing values of the [Formula: see text] and the magnetic number M. The thermal profile increases with higher [Formula: see text] and [Formula: see text] The concentration profile decreases with higher values of [Formula: see text] and [Formula: see text].

The technological regime and barriers to entry

Abstract The impact of the technological regime, defined in terms of appropriability, cumulativeness, and opportunity conditions on firms, has been examined in multiple studies. Within the analysis of technological regimes, firm entry has been of consistent interest due to its implications on competition and how firms may decide to enter an industry. However, studies show that two sources of nonlinearities warrant closer inspection, which could be achieved by looking at the entry of individual firms. The first is the interaction effects of the various dimensions of technological regimes. Their relevance is supported by both theoretical and empirical evidence, the analysis of which could be valuable for identifying the mechanisms behind the relationship between the technological regime and firm entry. Another is the role heterogeneity among entrants plays on the impact of the technological regime on firm entry, which has been found to be an important factor in many other aspects of firm behavior. This study thus aims to examine these two sources of nonlinearities, which can have significant implications for both firm strategy and policy.

Effects of the Seal Wire on the Nonlinear Dynamics of the Aircraft Engine Turbine Blades

Abstract Complicated systems made of multiple components are known to be difficult to model, considering their solutions can change dramatically even with the slightest variations in conditions. Aircraft engines contain such complicated systems, and some components in aircraft engines' turbines can cause significant changes in the system's overall response. Hence, this study is focused on investigating the behavior of a turbine blade of an aircraft engine and the effects of the contact between the blade and the seal wire on the dynamics of the blade-disk system. The investigation is performed via various numerical simulations in time and frequency domains. One sector of the bladed disk is modeled using the finite element method with the lock plate and the seal wire imposing cyclic symmetry boundary conditions in the static, modal, and frequency domain forced response analyses. In time domain analyses, the cyclic symmetry is replaced with simplified displacement restricting boundary conditions. The time domain analysis contains steady-state forced responses of the system. The results show that contact with the seal wire is not a major source of nonlinearity and damping. The contacts with the lock plate contribute more to the vibration damping than the seal wire. However, compared to the contacts at the root of the blade, both components remain less significant with regard to frictional damping and nonlinearity.