Nonlinear Case Research Articles

Generalisation of the hydrodynamics model method for hot and cold strip rolling application

The present work constitutes a generalization of the hydrodynamic model used to predict the pressures and the rolling speeds during the hot rolling of aluminum and copper strips. The hydrodynamic model with a linear behavior (Newton viscous) of the materials shows good predictions in the literature but its applicability is questionable in non-linear cases, when the materials exhibit viscoplastic or plastic behavior. This work extends the model to accommodate non-linear cases commonly encountered in rolling models (viscoplastic and plastic behaviors). The obtained results are in good agreement with experimental data from the literature. The validated model can, therefore, be considered an enhanced hydrodynamic model for predicting pressures and velocities during both hot and cold rolling of thin strips.

Modeling of railway track integrating a shock-absorbing mat at the sleepers-ballast interface using eigenfunctions of displacement

Damage caused to railway tracks and the surrounding environment during trains passage have become a real concern for those involved in the railway sector over the years. To find a solution for this problem, several approaches have already been implemented on railway tracks in some countries. Despite the efforts made in this area, the problem of ground vibration induced by railway traffic still remains a concern. Therefore, many studies have been focused on solutions to attenuate, even cancel vibrations effects on the track and nearby environment. This work has presented a modeling of a ballasted railway track with a shock-absorbing mat or Under Sleepers Rubber Mat (USRm) at the sleepers/ ballast interface. Using the equation of wave propagation in the ground and the constitutive law, the eigenfunctions of the displacement vector were determined. By coupling the equations of the track and those of the ground, the displacement, the velocity, and the acceleration of each track component, as well as the stress on the ground surface were expressed. The Newmark numerical method was used to represent these variables and to highlight the (USRm) influence on the dynamic behavior of the different track component. Comparison of the results obtained with the proposed railway model shows a reduction in vibration amplitudes of 64.9% and 94%for the rail, 86.5% and 56.25% for the sleepers. At the ground surface, the attenuation of vibration is evaluated at 53% and 6.85%, respectively for linear and nonlinear cases. The insertion of the shock-absorbing mat (USRm) in the railway track structure also reduces stress at the ground surface. The configuration of the railway track with a damping mat (USRm) therefore significantly reduces the transmitted vibration to the ground and to the nearby structures. It also contributes to the stability of the track and the improvement of its performance. The proposed railway track model with a shock-absorbing mat can be considered as a solution to reduce vibrations generated on track and the ground during trains passages.

Using Artificial Neural Networks to Solve the Gross–Pitaevskii Equation

The current work proposes the incorporation of an artificial neural network to solve the Gross–Pitaevskii equation (GPE) efficiently, using a few realistic external potentials. With the assistance of neural networks, a model is formed that is capable of solving this equation. The adaptation of the parameters for the constructed model is performed using some evolutionary techniques, such as genetic algorithms and particle swarm optimization. The proposed model is used to solve the GPE for the linear case (γ=0) and the nonlinear case (γ≠0), where γ is the nonlinearity parameter in GPE. The results are close to the reported results regarding the behavior and the amplitudes of the wavefunctions.

Three-Dimensional Site Response Analysis of Clay Soil Considering the Effects of Soil Behavior and Type

To understand changes in bedrock motion at the ground surface, frequency effects, and spatial distribution within the soil, it is important to look at how a site responds to earthquakes. This is important for soil–structure interaction in structural and geotechnical earthquake engineering. This study deals with the effect of classifying clays according to shear wave velocity (stiff/medium/soft) and nonlinearity in behavior (linear/nonlinear) on the analysis of the site response. A 3D soil model with a combination of free fields and quiet boundaries and advanced constitutive models for soil to obtain accurate results was used to conduct this study. A strong TABAS earthquake was used to excite the compliant base of the model after converting the velocity record of TABAS to an equivalent surface traction force using a horizontal force–time history proportional to the velocity–time history. This study reveals that the site response analysis is affected by the type of clay soil and the soil material behavior, with soft clay soil causing higher PGV and PGV values in the linear case and lower values in the nonlinear case due to soil yielding, which causes soil response attenuation. This results in extremely conservative and expensive building designs when linear soil behavior is adopted. On the other hand, the applied earthquake exhibits greater attenuation at longer frequencies and greater amplification at mid and short frequencies. However, at frequencies near the applied earthquake frequency, neither attenuation nor amplification occurs. Furthermore, nonlinear soil behavior is crucial for soil evaluation and foundation design due to higher octahedral shear strain and settlement values, especially in softer soils, resulting from extensive plastic deformation.

MIMO system identification and uncertainty calibration with a limited amount of data using transfer learning

Multiple-input multiple-output (MIMO) systems are fundamental in numerous advanced engineering applications, from aerospace to telecommunications, where precise system identification is critical for optimal performance. However, the identification of such systems often faces significant hurdles due to data scarcity, with existing approaches typically requiring substantial amounts of data for effective training. Addressing this challenge, this paper introduces a novel transfer learning framework designed specifically for MIMO system identification under conditions of limited data and inherent uncertainties. The proposed framework is applied to two case studies: the first in metal additive manufacturing, specifically the laser-blown powder-directed energy deposition as the source domain and the laser hot wire-directed energy deposition as the target domain, and the second involving a nonlinear case study of a continuous stirred-tank reactor (CSTR) with a temperature-dependent reaction. The results underscore the framework's effectiveness in capturing the dynamics of the target systems, including the ability to effectively model nonlinear dynamics. Comparative analyses highlight the benefits of employing dimensionless numbers in dynamic system modelling, offering reduced dimensionality, more physical meaning, and increased model accuracy. Overall, the proposed framework presents a promising approach to enhance system identification in MIMO systems with limited data and uncertainties, with potential applications across diverse domains.

Multi-instance learning in the presence of positive and unlabeled bags

Multi-instance learning is a classic algorithm within the weakly supervised learning framework. Most previous approaches to multi-instance learning have typically assumed that every bag comes with its own label. However, in real-world applications, obtaining fully-labeled bags can be a challenging and resource-intensive task, primarily due to the substantial time and labor required for labeling each instance. Fortunately, collecting bags with only positive and unlabeled instances is a more feasible option. In this paper, we aim to learn an efficient binary classifier using positive and unlabeled bags. To address the absence of negative bags, we adopt a novel strategy that combines label noise learning and multi-instance kernels. Our approach involves a comprehensive analysis of empirical risk minimization in the presence of label noise, allowing us to decompose the loss function for negative bags with false negative labels into two distinct parts. Importantly, only the second part of this loss function is affected by the presence of noisy labels. To mitigate the influence of noisy labels on the negative bags, we propose the Multi-Instance Kernel-based Centroid Estimation (MIKCE) technique. MIKCE is a versatile method that can be employed with various types of data, including linear and nonlinear cases. Additionally, we provide an upper bound on the generalization error, ensuring the convergence of the MIKCE algorithm. Finally, we conduct numerical experiments on 31 public datasets to empirically validate the effectiveness of the MIKCE approach.

Critical exponent to a cancer invasion model with nonlinear diffusion

This paper is concerned with a cancer invasion model that incorporates porous medium diffusion (Δum) and extracellular matrix remodeling effects [ηω(1 − u − ω)] in a bounded domain of RN (N ≥ 2). Rich achievements have been achieved for the case η = 0 in the past ten years for the nonlinear diffusion case, but there is no any progress for η > 0. In this paper, we pay our attention to the global existence of solutions of the case η > 0, and establish the critical exponent m*=2N−2N of global solvability. More precisely, if m > m*, the solution will always exist globally, while if m < m*, there exist blow-up solutions. In this system, the remodeling effect of extracellular matrix [ηω(1 − u − ω)] bring some essential difficulties to the estimation of the haptotactic term, so the main technique we used is completely different from the case of η = 0.

Stability of KdV equation on a network with bounded and unbounded lengths

In this work, we studied the exponential stability of the nonlinear KdV equation posed on a finite star shaped network with finite number of branches. On each branch of the network we define a KdV equation posed on a finite domain [[EQUATION]] or the half-line [[EQUATION]]. We start by proving well-posedness and some regularity results. Then, we state the exponential stability of the linear KdV equation by acting with a damping term on some branches. The main idea is to prove a suitable observability inequality. In the nonlinear case, we obtain two kinds of results: The first result holds for small amplitude solutions, and is proved using a perturbation argument from the linear case but without acting on all edges. The second result is a semiglobal stability result, and it is obtained by proving an observability inequality directly for the nonlinear system, but we need to act with damping terms on all the branches. In this case, we are able to prove the stabilization in weighted spaces.

Optimization of calculations in modeling the breaking of plasma oscillations

The proposed method of optimization of calculations is of a combined nature. First, using a piecewise uniform grid, a difference scheme of the second order of accuracy of the McCormack type is constructed, and then — to weaken the stability condition — an implicit version of the scheme is used, which even in the nonlinear case allows an iteration-free implementation. When constructing a grid, a preliminary study is required, which is based on the analytical properties of the solution of the differential problem. Based on the calculations carried out, it can be concluded that the potential reduction in the amount of calculations is dozens of times without a noticeable loss of accuracy.

A high-static-low-dynamic-stiffness delayed resonator vibration absorber

Delayed resonator (DR), which enables complete vibration suppression through loop delay tuning, has been extensively investigated as a linear active dynamic vibration absorber since its invention. Besides, the nonlinear high-static-low-dynamic stiffness (HSLDS) has been widely used in vibration isolators for broadband (yet incomplete) vibration reduction. This work combines the benefits of DR and the nonlinear HSLDS characteristics, thus creating a nonlinear DR (NDR). Three unexplored topics are considered: (1). To address the nonlinear dynamics that are made complicated by the introduction of delay and the nonlinearity coupled between the NDR and the primary structure; (2). To tune the control parameters to seek possible complete vibration suppression in the nonlinear case, and accordingly, to determine the operable frequency band; (3). To evaluate how the HSLDS characteristics can enhance the performance of the linear DR (LDR) and how to design an NDR to maximize its benefits. Without loss of generality, a classic nonlinear HSLDS structure with three springs and two links is considered, and mathematical tools and computational algorithms are introduced or proposed to efficiently address theoretical analysis. Using the parameters of an experimental setup, we show that a properly tuned NDR suppresses the vibrations on the primary structure to a sufficiently low level. Besides, the HSLDS characteristics extend the operable frequency band compared with the LDR, while strong nonlinearity limits such extension. The proposed analysis procedures for delay-coupled nonlinear dynamics, alongside the control parameter tuning, determination of operable frequency bands, and structural design rules, establish a basic theoretical framework for the NDR design.

Refined Reservoir Routing (RRR) and Its Application to Atmospheric Carbon Dioxide Balance

Reservoir routing has been a routine procedure in hydrology, hydraulics and water management. It is typically based on the mass balance (continuity equation) and a conceptual equation relating storage and outflow. If the latter is linear, then there exists an analytical solution of the resulting differential equation, which can directly be utilized to find the outflow from known inflow and to obtain macroscopic characteristics of the process, such as response and residence times, and their distribution functions. Here we refine the reservoir routing framework and extend it to find approximate solutions for nonlinear cases. The proposed framework can also be useful for climatic tasks, such as describing the mass balance of atmospheric carbon dioxide and determining characteristic residence times, which have been an issue of controversy. Application of the theoretical framework results in excellent agreement with real-world data. In this manner, we easily quantify the atmospheric carbon exchanges and obtain reliable and intuitive results, without the need to resort to complex climate models. The mean residence time of atmospheric carbon dioxide turns out to be about four years, and the response time is smaller than that, thus opposing the much longer mainstream estimates.

Fluctuation effects in superconducting nanowires under electric field

This study applies the time-dependent Ginzburg–Landau (TDGL) theory with thermal noise to analyze the thermoelectric and transport properties of superconducting Sn nanowires, focusing on the thermopower (αxx) and the phase transition characteristics in the S-shaped J-E curves. We observe significant contributions from superconducting Cooper pairs, which remain nonzero above the critical temperature (Tc), indicating residual superconductivity due to strong thermal fluctuations. The S-like shape in the J-E curves is attributed to a dynamical instability transition temperature (T∗) at approximately 2.9 K, where thermal fluctuations dominate. Furthermore, we compare the resistance in the linear response to experimental data for Sn nanowires both below and above Tc. Below Tc, the resistance sharply decreases, reflecting the robust superconducting state, while above Tc, it increases, aligning with normal state behavior. In nonlinear response case, our results indicate that high electric fields can be effectively used to suppress order-parameter fluctuations and the electrical conductivity in superconducting nanowires. The findings provide critical insights into the thermoelectric behavior and phase transitions in Sn nanowires, highlighting the importance of Cooper pair dynamics in shaping the transport properties of one-dimensional superconductors. This understanding is essential for the development of advanced nanoelectronic devices leveraging these unique superconducting properties.

The Comparison A-Optimal and I-Optimal Design in Non-Linear Models to Increase Purity Levels Silicon Dioxide

One of the obstacles that arise in optimal design is the non-linear model. The relationship between temperature factors and the temperature increase rates with the purity of silicon dioxide (SiO2) forms a non-linear pattern. Determining the optimal design for a non-linear model is relatively more complex than a linear model because it requires additional information in its information matrix. Therefore, this issue necessitates further research on optimal design in non-linear models. This study uses the polynomial Taylor approach to approximate the non-linear equation through a linear equation using the appropriate optimal design methods, namely A-Optimal and I-Optimal criterion. The point search algorithm used was variable neighborhood search, this algorithm searches for design points by exploring several different neighborhood structures. These two methods were chosen to compare the characteristics and performance of the designs produced, aiming to obtain an optimal design to improve SiO2 purity (non-linear case) using the same algorithm, VNS. The research results showed that the design pattern produced by the A-Optimal design formed three temperature groups, namely the minimum temperature of 800°C - 820°C, the middle temperature of 850°C, 860°C, and the maximum temperature of 900°C, with varying temperature increase rates in the design area. The design pattern produced by the I-Optimal design formed a full quadratic pattern, namely the minimum temperature of 800°C and the maximum temperature of 900°C, with varying temperature increase rates in the design area. The I-Optimal design demonstrated the best performance (most optimal) in the aspect of prediction variance compared to the A-Optimal design across all alternative points in this study to improve SiO2 purity.

Transient modes for the coupled modified Korteweg-de Vries equations with negative cubic nonlinearity: Stability and applications of breathers.

Dynamics and properties of breathers for the modified Korteweg-de Vries equations with negative cubic nonlinearities are studied. While breathers and rogue waves are absent in a single component waveguide for the negative nonlinearity case, coupling can induce regimes of modulation instabilities. Such instabilities are correlated with the existence of rogue waves and breathers. Similar scenarios have been demonstrated previously for coupled systems of nonlinear Schrödinger and Hirota equations. Both real- and complex-valued modified Korteweg-de Vries equations will be treated, which are applicable to stratified fluids and optical waveguides, respectively. One special family of breathers for coupled, complex-valued equations is derived analytically. Robustness and stability of breathers are studied computationally. Knowledge of the growth rates of modulation instability of plane waves provides an instructive prelude on the robustness of breathers to deterministic perturbations. A theoretical formulation of the linear instability of breathers will involve differential equations with periodic coefficient, i.e., a Floquet analysis. Breathers associated with larger eigenvalues of the monodromy matrix tend to suffer greater instability and increased tendency of distortion. Predictions based on modulation instability and Floquet analysis show excellent agreements. The same trend is obtained for simulations conducted with random noise disturbances. Linear approaches like modulation instabilities and Floquet analysis, thus, generate a very illuminating picture of the nonlinear dynamics.

Experimental analysis of extreme wave loads on a containership

Model-scale experiments were conducted to investigate the non-linearities associated with the Vertical Bending Moment (VBM) of a rigid container vessel in irregular waves. Long-duration runs ( ≃ 25 hours of data in full scale) were performed to allow the calculation of statistically meaningful extreme values of hogging and sagging on five different sea-states (from Hs = 6 m up to Hs = 17 m).The model tests are presented in detail, and sampling errors are estimated. It is found that the ratio between linear and non-linear VBM (the so-called non-linear factor) can be modeled as a function of the linear value only. For a given linear value, the non-linear correction is the same, whatever the sea state. This observation, new in the fully non-linear case, is very valuable as it opens the door to efficient design assessment methodologies such as increased design sea state or design waves.

A Numerical Method for Investigating Fractionali Volterra-Fredholm Integro-Differential Model

In this article, we investigate the fractional Volterra-Fredholm integro-differential equations. These equations appear in several applications such as control theory, biology, and particle dynamics in physics. We derive a numerical method based on the operational matrix method to solve this class of integro-differential equations. We prove the existence and uniqueness of the ex-act solution. Additionally, we demonstrate the uniform convergence of the numerical solutions to the exact solution. We present several numerical examples to show the numerical efficiency of the proposed method. In the first example, we choose a linear problem and find that the approximate solution converges to the exact solution when the number of block pulse functions is very large. In the next two examples, we consider the nonlinear case and compute the L2-local truncation error since exact solutions are not available. The error was of order 10−12. Furthermore, we sketch the graph of the approximate solutions for different values of the fractional derivative to observe the influence of the fractional derivative on the profile of the solutions. Theoretical and numerical results show that the proposed method is accurate and can be applied to other nonlinear problems in science.

Gaussian process regression driven rapid life-cycle based seismic fragility and risk assessment of laminated rubber bearings supported highway bridges subjected to multiple uncertainty sources

Structural attributes, corrosion of steel rebars in reinforced concrete (RC) columns, thermal oxidative aging of inherent rubber in laminated rubber bearings (LRBs), replacement of LRBs, and the time-dependent seismic hazard level at the bridge site are well recognized as the main factors that affect the accurate assessment of the seismic performance of RC bridges implemented with LRBs. However, the probabilistic seismic demand model (PSDM) and probabilistic structural capacity model (PSCM) of deteriorated bridge components may not exhibit homoscedastic log-linearity. And the expansion of seismic fragility surfaces and seismic risk curves in the temporal dimension requires a considerable computational cost on the nonlinear analysis, which regretfully has not been well addressed in current studies. To fill in this gap, this study proposes a life-cycle based seismic fragility and risk assessment framework for LRBs supported RC bridges considering deterioration mechanisms of bridge components during their life cycles demonstrated by a three-span LRBs supported highway bridge considering multiple uncertainties such as the replacement interval for LRBs. In this framework, Gaussian process regression (GPR) is employed to effectively construct the PSDM and PSCM of deteriorated bridge components, and then the time-dependent seismic fragility surfaces of bridge components and bridge system are established. Life-cycle based seismic risk analysis is conducted to quantify the coupled time-dependent effects of deterioration and site seismic hazard on a newly built bridge during its design service life and a deteriorated bridge within its remaining service life, respectively. Results indicate that GPR can successfully capture the nonlinearity and heteroscedasticity characteristics of PSDM and PSCM, leading to a significant reduced number of nonlinear analysis cases to generate the time-dependent seismic fragility surfaces and seismic risk curves. Consequently, the proposed framework using GPR can significantly improve the accuracy and efficiency of life-cycle based seismic performance assessment when uncertainties of structural attributes, deterioration progress and seismic hazard are taken into consideration.

Application of single surface isotropic damage plasticity model in nonlinear dynamic analysis of the Koyna Dam

PurposeIn the present study, the application of a recent damage plasticity model is presented for nonlinear dynamic analysis of the Koyna gravity dam. This is a single surface isotropic damage plasticity concrete model, which is based on the decomposition of stresses and was proposed in a previous study. The theoretical aspects of the model are initially reviewed, and a few preliminary verification examples are illustrated. Thereafter, the HHT-α (i.e. Hilber–Hughes–Taylor) algorithm is presented for nonlinear dynamic analysis of concrete gravity dams.Design/methodology/approachBased on the prepared tools, nonlinear behavior of the Koyna Dam is studied by applying the invoked damage plasticity model. For this purpose, three cases are considered for the present study. Case A, which is based on the linear model, is mainly used for comparative purposes. The other two cases (B and C) correspond to the nonlinear (i.e. damage plasticity) model. The basic data for these two cases are similar. However, the employed damping algorithms are different and correspond to constant and variable damping algorithms, respectively. This means that the damping matrix is either kept constant or updated for all iterations of different time increments through the course of analysis.FindingsThe time histories of horizontal displacement at the dam crest were initially compared for the three cases: the linear Case A, and two nonlinear Cases B and C. It was observed that nonlinear cases’ responses begin to deviate from the corresponding linear case after the time of about 4.3 s. However, the amount of change for Case C (i.e. variable damping) was much greater than for Case B (i.e. constant damping). This was manifested initially in the peaks of response. It was also noticed that the period of response changed slightly for Case B in comparison with the linear Case A, while this change was significant for Case C. The obtained tensile and compressive damages were subsequently compared for the two nonlinear cases. For constant damping Case B, it was noticed that tensile damage occurred in the D/S face kink and on the U/S face slightly at a lower elevation. Moreover, it had a scattered nature. However, in variable damping Case C, it was noticed that tensile damage was much more localized and acted similar to a discrete crack. Of course, both cases also show tensile damages at the dam’s heel. In regard to compressive damages, it is observed that low values are occurring for both nonlinear cases as expected.Originality/valueThe application of a recent single surface isotropic damage plasticity concrete model is presented for nonlinear dynamic analysis of the Koyna gravity dam. The nonlinear response of the dam is investigated for two different damping algorithms. Moreover, the influence of variable characteristic length is also investigated in the latter part of this study.

Spectral deferred correction method for fractional initial value problem with Caputo–Hadamard derivative

This paper considers an efficient and accurate spectral deferred correction (SDC) method for the initial value problem (IVP) with Caputo–Hadamard derivative. We first apply the basic idea of the SDC method to derive the numerical scheme. Then the iteration matrix which is the key to convergence of the proposed scheme can be obtained for the linear problem. Detailed computation of history term is presented using the spectral collocation method based on mapped Jacobi log orthogonal functions (MJLOFs). Finally, numerical simulations for both linear and nonlinear cases are shown to verify the feasibility and efficiency of the proposed method.

Neural Networks and the Nonlinear Feynman–Kac Theorem Applied to Financial Options Pricing

The classic financial market model proposed by Black and Scholes allows solving, under a series of simplifying assumptions, the problem of valuing financial derivatives. However, these assumptions deviate from the reality of global markets, particularly assuming perfect liquidity. In this work, a financial market model is proposed in which the illiquidity of the risky asset is stochastic, described by a mean-reversion process. The dynamics of the asset price in this context are established alongside the valuation PDE for derivatives. Additionally, as an approximation method to the solution of this PDE, the implementation of the extension of the Feynman–Kac representation theorem to the nonlinear case using deep neural networks is proposed.