Nonlinear Limit Research Articles

On Ill‐Conditioning of Descent Methods to Recover Initial Data for the Burgers Equation With Small Viscosity

ABSTRACTThe identification of the initial condition of a parabolic equation given terminal data is a problem arising in several applications. In this work, we consider the problem for the semilinear parabolic viscous Burgers equation in the vanishing viscosity case. The limiting equation is the nonlinear hyperbolic Burgers equation, for which there is non‐uniqueness backward in time. In both cases, solutions are sought by formulating the identification problem as the minimization of a least squares functional, and gradient descent methods are used for computation. Following this approach, we construct the first and second variations of the functional by the adjoint state approach. We prove that the second variation is continuous with respect to viscosity. Consequently, gradient descent methods of approximation inherit the ill‐conditioning of the nonlinear hyperbolic limit.

Evaluating a double integral using Euler's method and Richardson extrapolation

We transform a double integral into a second-order initial value problem, which we solve using Euler's method and Richardson extrapolation. For an example we consider, we achieve accuracy close to machine precision (~10-13). We find that the algorithm is capable of determining the error curve for an arbitrary cubature formula, and we use this feature to determine the error curve for a Simpson cubature rule. We also provide a generalization of the method to the case of nonlinear limits in the outer integral.

Mechanism analysis on a type of limit induced oscillation for single‐machine infinite‐bus system of PMSG with positive damping

AbstractThe mechanism of oscillations that occur in the system connected to renewable energy sources is complex, generally including negative damping oscillation, limit induced oscillation etc. Combining with the dynamic characteristics of the low‐dimensional system after limit working, this paper analyses the mechanism of a type of limit induced oscillation that occurs when the output of the DC voltage outer loop continuously touches the limit for single‐machine infinite‐bus system of permanent magnet synchronous generator with positive damping under a large disturbance. Specific study as follows: firstly, a non‐smooth state space model of single‐machine infinite‐bus system of permanent magnet synchronous generator with nonlinear state limit (clarified as a Filippov system) is established. Secondly, the small disturbance characteristics of the system are analysed when the limit does not work. Thirdly, the piecewise dynamic description of the system is analysed with the limit working, and the reason that the final contraction of the system to an eight‐dimensional manifold is explained based on mathematical derivation and physical characteristics. Finally, the characteristics of equilibrium point of the system in a low‐dimensional manifold are analysed when the limit continuously working, revealing that the existence of a pair of conjugate unstable eigenvalues in the low‐dimensional system is the reason of system oscillation.

New adaptive low-dissipation central-upwind schemes

We introduce new second-order adaptive low-dissipation central-upwind (LDCU) schemes for the one- and two-dimensional hyperbolic systems of conservation laws. The new adaptive LDCU schemes employ the recently proposed LDCU numerical fluxes computed using the point values reconstructed with the help of adaptively selected nonlinear limiters. To this end, we use a smoothness indicator to detect “rough” parts of the computed solution, where the piecewise linear reconstruction is performed using an overcompressive limiter, which leads to extremely sharp resolution of shock and contact waves. In the “smooth” areas, we use a more dissipative limiter to prevent appearance of artificial kinks and staircase-like structures there. In order to avoid oscillations, we perform the reconstruction in the local characteristic variables obtained using the local characteristic decomposition. We use a smoothness indicator from Löhner (1987) [34] and apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics. The obtained numerical results clearly demonstrate that the new adaptive LDCU schemes outperform the original ones.

Suppression Limit Cycles in 2x2 Nonlinear Systems with Memory Type Nonlinearities

For several decades, the importance and weight-age of prediction of nonlinear self-sustained oscillations or Limit Cycles (LC) and their quenching by signal stabilization have been discussed, which is confined to Single Input and Single Output (SISO) systems. However, for the last five to six decades, the analysis of 2x2 Multi Input and Multi Output (MIMO) Nonlinear Systems gained importance in which a lot of literature available. In recent days’ people have started discussing suppression of LC which limits the performance of most of the physical systems in the world. It is a formidable task to suppress the limit cycles for 2x2 systems with memory type nonlinearity in particular. Backlash is one of the nonlinearities commonly occurring in physical systems that limit the performance of speed and position control in robotics, automation industry and other occasions like Load Frequency Control (LFC) in multi area power systems. The feasibility of suppression of such nonlinear self-oscillations has been explored in case of the memory type nonlinearities. Backlash is a common memory type nonlinearity which is an inherent Characteristic of a Governor, used for usual load frequency control of an inter-connected power system and elsewhere. Suppression LC using pole placement technique through arbitrary selection and optimal selection of feedback Gain Matrix K with complete state controllability condition and Riccati Equation respectively and is done through state feedback. The Governing equation is d/dt [X(t)] =(A-BK) X: which facilitates the determination of feedback gain matrix K for close loop Poles / Eigen values placement where the limit cycles are suppressed/eliminated in the general multi variable systems. The complexity involved in implicit non-memory type or memory type nonlinearities, it is extremely difficult to formulate the problem for 2x2 systems. Under this circumstance, the harmonic linearization/harmonic balance reduces the complexity considerably. Still the analytical expressions are so complex which loses the insight into the problem particularly for memory type nonlinearity in 2x2 system and the method is made further simpler assuming a 2x2 system exhibits the LC predominately at a single frequency. Hence in the proposed work an alternative attempt has been made to develop a graphical method for the prediction of Limit Cycling Oscillations in 2x2 memory type Nonlinear systems which not only reduces the complexity of formulations but also facilitates clear insight into the problem and its solution. The present techniques are well illustrated with an example and validated / substantiated by digital simulation (developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software. The present work has the brighter future scope of: Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional nonlinear systems through an exhaustive analysis. Analytical/Mathematical methods may also be developed for signal stabilization using both deterministic and random signals based on Dual Input Describing function (DIDF) and Random Input Describing Function (RIDF) respectively. The phenomena of Synchronization and De-synchronization can be observed/identified analytically using Incremental Input Describing Function (IDF), which can also be validated by digital simulations.

Quenching and Suppression of Limit Cycles in 3x3 Nonlinear Systems

For several decades, the importance and weight-age of prediction of nonlinear self-sustained oscillations or Limit Cycles (LC) and their quenching by signal stabilization have been discussed which is confined to Single Input and Single Output (SISO) system. However, for the last five to six decades, the analysis of 2x2 Multi Input and Multi Output (MIMO) Nonlinear Systems gained importance in which a lot of literature available. In recent days few literatures are available which addresses the exhibition of LC and their quenching/suppression in 3x3 MIMO Nonlinear systems. Poor performances in many cases like Load Frequency Control (LFC) in multi area power system, speed and position control in robotics, automation industry and other occasions have been observed which draws attention of Researchers. The complexity involved, in implicit nonmemory type and memory type nonlinearities, it is extremely difficult to formulate the problem in particular for 3x3 systems. Under this circumstance, the harmonic linearization/ harmonic balance reduces the complexity considerably. Still the analytical expressions are so complex which loses the insight into the problem particularly for memory type nonlinearity in 3x3 system. Hence in the present work a novel graphical method has been developed for prediction of limit cycling oscillations in a 3x3 nonlinear system. The quenching of such LC using signal stabilization technique using deterministic (Sinusoidal) and random (Gaussian) signals has been explored. Suppression LC using pole placement technique through arbitrary selection and optimal selection of feedback Gain Matrix K with complete state controllability condition and Riccati Equation respectively. The method is made further simpler assuming a 3x3 system exhibits the LC predominantly at a single frequency, which facilitates clear insight into the problem and its solution. The proposed techniques are well illustrated with example and validated/substantiated by digital simulation (a developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software. The Signal stabilization with Random (Gaussian) Signals and Suppression LC with optimal selection of state feedback matrix K using Riccati Equation for 3x3 nonlinear systems have never been discussed elsewhere and hence it claims originality and novelty. The present work has the brighter future scope of: i. Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional nonlinear systems through an exhaustive analysis. ii. Analytical/Mathematical method may also be developed for signal stabilization using both deterministic and random signals based on Dual Input Describing function (DIDF) and Random Input Describing Function (RIDF) respectively. iii. The phenomena of Synchronization and De-synchronization can be observed/identified analytically using Incremental Input Describing Function (IDF), which can also be validated by digital simulations.

Some remarks on the effect of the Random Batch Method on phase transition

In this article, we focus on two toy models : the Curie–Weiss model and the system of N particles in linear interactions in a double well confining potential. Both models, which have been extensively studied, describe a large system of particles with a mean-field limit that admits a phase transition. We are concerned with the numerical simulation of these particle systems. To deal with the quadratic complexity of the numerical scheme, corresponding to the computation of the O(N2) interactions per time step, the Random Batch Method (RBM) has been suggested. It consists in randomly (and uniformly) dividing the particles into batches of size p>1, and computing the interactions only within each batch, thus reducing the numerical complexity to O(Np) per time step. The convergence of this numerical method has been proved in other works.This work is motivated by the observation that the RBM, via the random constructions of batches, artificially adds noise to the particle system. The goal of this article is to study the effect of this added noise on the phase transition of the nonlinear limit, and more precisely we study the effective dynamics of the two models to show how a phase transition may still be observed with the RBM but at a lower critical temperature.

Dislocation-based finite element method for homogenized limit domain characterization of structured metamaterials

PurposeHeterogeneous and micro-structured materials have been the object of multiscale and homogenization techniques aimed at recognizing the elastic properties of the equivalent continuum. The proposed investigation deals with the mechanical characterization of the heterogeneous material structured metamaterials through analyzing the ultimate strength using the limit analysis of the Representative Volume Element (RVE). To get the desired material strength, a novel finite element formulation based on the derivation of self-equilibrated solutions through the finite elements devoted to calculating the lower bound theorem has been implemented together with the limit analysis in Melàn’s formulation.Design/methodology/approachThe finite element formulation is based on discrete mapping of Volterra dislocations in the structure using isoparametric representation. Using standard finite element techniques, the linear operator V, which relates the self-equilibrated internal solicitation to displacement-like nodal parameters, has been built through finite element discretization of displacement and strain.FindingsThe proposed work presented an elastic homogenization of the mechanical properties of an elementary cell with a geometry known in the literature, the isotropic truss. The matrix of elastic constants was calculated by subjecting the RVE to numerical load tests, simulated with a commercial FEM calculation code. This step showed the dependence of the isotropy properties, verified with Zener theory, on the density of the RVE. The isotropy condition of the material is only achieved for certain section ratios between body-centered cubic (BCC) and face-centered cubic (FCC), neglecting flexural effects at the nodes. The density that satisfies Zener’s conditions represents the isotropic geomatics of the isotropic truss.Originality/valueFor the isotropic case, the VFEM procedure was used to evaluate the isotropy of the limit domain and was compared with the Mises–Schleicher limit domain. The evaluation of residual ductility and dissipation energy allowed a measurement parameter for the limit anisotropy to be defined. The novelty of the proposal consisted in the formulation of both the linearized and the nonlinear limit locus of the material; hence, it furnished the starting point for further limit analysis of the structures whose elementary volume has been described through the proposed approach.

Effect of morphology on the optical limiting of BiVO4 films

Precise control over the morphology and crystal structure of semiconductor-based optical materials is essential for optimizing their nonlinear optical properties. Morphological factors significantly affect the material’s interaction with light, influencing nonlinear absorption, harmonic generation, and optical confinement. In this study, the nonlinear optical (NLO) properties and optical limiting (OL) performance of bismuth vanadate (BiVO4) films depending on precursor solution pH condition, powder morphology and weight percentage of the BiVO4 powders in PMMA (10–30 wt.%) were investigated. The BiVO4 powders were synthesized via the hydrothermal method and the effect of precursor solution pH (pH of 1, 7.5 and 12) was investigated on the size and morphology of the BiVO4 powders. BiVO4 films were prepared using the spin coating method utilizing the BiVO4 powders and PMMA. The bandgap values of films were found to change from 3.39 to 3.56 eV depending on the BiVO4 growth solution pH. The nonlinear absorption coefficients (βeff), saturation intensity thresholds (ISAT) and optical limiting (OL) threshold were obtained using the open aperture (OA) Z-scan experiment. It has been observed that the nonlinear absorption and optical limitation performance of the BiVO4 films can be altered by changing the powder morphology and BiVO4 weight concentration. BiVO4 thin films fabricated using a precursor solution pH 12 and weight concentration of 30 wt.% showed the best performance among all fabricated films with the highest nonlinear absorption coefficient (βeff), saturation density threshold (ISAT) and lowest OL threshold value.

Recirculation through western boundary currents varies nonlinearly with the ocean basin's aspect ratio

Recirculation gyres adjacent to western boundary currents (WBCs) in the ocean enhance the poleward transport of these currents. While it is well-established that the WBC in a barotropic ocean strengthens with increase in basin's aspect ratio (the meridional-to-zonal extent ratio), how intensity of the recirculation through the western boundary layer varies with this parameter remains unexplored. I address this using the non-dimensional form of the nonlinear, wind-driven Stommel–Munk model of westward intensification that comprises three parameters—the aspect ratio (δ), the damping coefficient (ϵ), and the β-Rossby number (Rβ). Here, ϵ is set by the ratio of Rayleigh friction coefficient (or eddy viscosity) to the meridional gradient of the Coriolis frequency and the basin's zonal dimension, while Rβ is proportional to wind stress amplitude and quantifies the strength of nonlinearity. In the weak-to-moderate nonlinearity limit (Rβ<∼ϵ), perturbation analysis reveals that recirculation varies concavely with aspect ratio, suggesting existence of an optimal aspect ratio (δopt) for which the recirculation is maximum and for typical values of ϵ (10−3−10−2), δopt follows the power-law relation δopt=4.3ϵ. Numerical simulations further validate the existence of δopt. For large ϵ (>5×10−3), the power-law predicts δopt for the numerical solutions rather accurately, but does not hold for smaller ϵ (2×10−3) due to increased importance of nonlinear terms. Nevertheless, the nonlinear variation in recirculation through the western boundary layer with aspect ratio is observed for all ϵ values and may contribute to the heterogeneous increase in the WBC's transport across different ocean basins in a warming climate.

Numerical study on bifurcation characteristics of wind-induced vibration for an H-shaped section

In order to reveal the influence of initial excitation on the bifurcation phenomenon of bridge decks, a new perspective of flow characteristics is developed based on the computational fluid dynamics numerical simulation method. Then, the bifurcation mechanism of vortex-induced vibration (VIV) response and nonlinear flutter response of the H-shaped section is investigated. The results show that when the wind speed is 2 m/s, under a small torsional excitation of 0.5°, the flow field of the H-shaped section will develop into the vortex shedding mode of the vertical vibration, resulting in vertical VIV. However, while under a large excitation of 6°, the flow field will directly transform into the vortex shedding mode of the torsional vibration, resulting in torsional VIV. Therefore, the bifurcation phenomenon of the VIV response is observed. When the wind speed is 4 m/s, the H-shaped section exhibits a nonlinear flutter limit cycle oscillation under a large excitation of 8°, but its response can be ignored under a small excitation of 0.5°. This phenomenon is attributed to the significant change in the transition of the vortex shedding mode from a small amplitude to a stable large amplitude, and the flow field lacks enough energy to complete the transition of the vortex shedding mode, resulting in the bifurcation phenomenon of the nonlinear flutter response. When the wind speed is 3.0 m/s, the large excitation will change the vortex shedding frequency of the new H-shaped section, resulting in the torsional VIV.

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.

Hamiltonian formalism for bistable-multilayered plates under non-mechanical stimuli

Bistable multilayered plates have attracted significant attention as morphing and adaptive structures, renowned for their unprecedented exceptional performance. However, accurately predicting both their shape and internal stresses poses formidable challenges due to their geometrically nonlinear nature. For the first time, this paper introduces a Hamiltonian formalism aimed at achieving high-resolution analysis of nonlinear multilayered plates subjected to non-mechanical stimuli, including hygro-thermo-electro-magneto-elastic responses. A canonical system is strategically developed to compute membrane behaviors, yielding symplectic dual differential equations for in-plane field variables. This method elegantly decouples out-of-plane variables from the full-state vector, leading to an exact analytical solution in the membrane problem. To predict the bending behaviors, the first variation of the Hamiltonian energy density function, expressed as a power series of the transverse deflection, ensures flexural equilibrium. The power series effectively captures all admissible out-of-plane deformations, enabling a smooth transition between linear and nonlinear plate responses, including pitchfork bifurcation and limit points. The validity and accuracy of the proposed method are rigorously assessed through convergence tests and satisfaction of boundary conditions across various equilibria, ranging from monostability to bistability. It is cross-verified with high-fidelity finite-element methods (FEM), showing excellent agreements in both deformations and stress resultants. This research presents a physics-based methodology that unveils a parametric interplay between in-plane and out-of-plane field variables, serving as an efficient approach for analyzing adaptive and morphing structures.

Linearity assessment: deviation from linearity and residual of linear regression approaches.

In this computer simulation study, we examine four different statistical approaches of linearity assessment, including two variants of deviation from linearity (individual (IDL) and averaged (AD)), along with detection capabilities of residuals of linear regression (individual and averaged). From the results of the simulation, the following broad suggestions are provided to laboratory practitioners when performing linearity assessment. A high imprecision can challenge linearity investigations by producing a high false positive rate or low power of detection. Therefore, the imprecision of the measurement procedure should be considered when interpreting linearity assessment results. In the presence of high imprecision, the results of linearity assessment should be interpreted with caution. Different linearity assessment approaches examined in this study performed well under different analytical scenarios. For optimal outcomes, a considered and tailored study design should be implemented. With the exception of specific scenarios, both ADL and IDL methods were suboptimal for the assessment of linearity compared. When imprecision is low (3 %), averaged residual of linear regression with triplicate measurements and a non-linearity acceptance limit of 5 % produces<5 % false positive rates and a high power for detection of non-linearity of>70 % across different types and degrees of non-linearity. Detection of departures from linearity are difficult to identify in practice and enhanced methods of detection need development.

Enhanced Nonlinear Optical Limiter in the Visible Spectral Region Based on Fe- and Co-Doped NiO Nanoparticles within PVP Nanofibers

Enhanced Nonlinear Optical Limiter in the Visible Spectral Region Based on Fe- and Co-Doped NiO Nanoparticles within PVP Nanofibers

Pointing correction based on limit cycles oscillators applied on a heliostats field digital twin

Digital twin models are an emerging topic that captures the dynamic response of physical systems to have a portrayal of digital representation. Complex systems such as heliostat fields for sustainable energy are good candidates for developing digital twins since the physical operation of facilities of this type can be highly time-consuming and weather-dependent. Therefore, a digital twin model of heliostat fields represents an adequate choice for the performance validation of new pointing correction algorithms with shorter assessing times. In this sense, one of the most important issues in the operation of heliostat fields for sustainable energy is to achieve a good accuracy of heliostats pointing to the target, even under physical errors of the solar field components and disturbances produced by external agents. However, since these types of disturbances have nonlinear characteristics, they should be dealt with by using nonlinear methods. This article presents a novel pointing correction technique based on nonlinear Limit Cycle Oscillators (LCOs) applied to the digital twin of a heliostat field. Analytical analysis, numerical verification, and further validation results are used to demonstrate the performance of the developed novel dynamic method.

In-flight characterization of a compact airborne quantum cascade laser absorption spectrometer

Abstract. Here, we report on the development of a new quantum cascade laser infrared absorption spectroscopy (QLAS) instrument, the Airborne Tropospheric Tracer In-situ Laser Absorption spectrometer (ATTILA), for atmospheric trace-gas measurements on board of the German High-Altitude Long-range Observatory (HALO) aircraft. Its small and light design makes it suitable for airborne measurements up to approximately 150 hPa of ambient pressure (13–14 km). The dual laser instrument can measure several trace gases simultaneously in two 36.4 m path astigmatic Herriott cells with a data acquisition frequency of 1 Hz. We describe the measurement method and the data acquisition of ATTILA and its in-flight performance by focusing on potential sources of influences on the signal, which we investigated with a dedicated test flight during which the instrument sampled from a constant source. We show that linear critical influences associated with challenging movement patterns can be corrected afterwards, while nonlinear limitations can be minimized by appropriate calibration frequencies and integrated time intervals. During the recent aircraft campaign CAFE Brazil (Chemistry of the Atmosphere Field Experiment in Brazil) from December 2022 to January 2023, carbon monoxide (CO) measurements from ATTILA show a good agreement of a R2 of 0.89 with simultaneous CO measurements from an established IR spectrometer for airborne measurements, the TRacer In Situ TDLAS (tunable diode laser absorption spectroscopy) for Atmospheric Research (TRISTAR), at a 10 s time resolution. First dynamical characteristics and tracer distributions of CO and methane (CH4) over the Amazon rainforest can be identified with ATTILA measurements with a total measurement uncertainty of 10.1 % and 17.5 % for calibration gas mixing ratios of 153 and 1990 ppbv and a data accuracy of 0.3 % and 5.5 % for a data acquisition frequency of 1 Hz for CO and CH4, respectively.

Reliability and sensitivity analysis of delamination growth of composite laminate structures using two efficient sampling methods

In recent years, composite structures have been used in a large number of applications in aerospace, machinery, marine, and civil engineering. However, there are inevitably many uncertainties in the whole life cycle of composite structures, which can easily lead to structural damage and failure. Therefore, it is important to analyze the reliability and sensitivity of composite structures. At present, most of the contributions use the first-order reliability method (FORM) and the second-order reliability method (SORM) to study the reliability of composite structures and compare them with the results of the Monte Carlo simulation. However, both methods have their limitations. FORM cannot guarantee the calculation accuracy for the highly nonlinear limit state equation, and the calculation efficiency of SORM is too low. Therefore, this paper proposes to use importance sampling (IS) and backpropagation neural network-based Monte Carlo (MC-BPNN) to study the reliability, sensitivity, and dispersion of delamination growth of composite laminates. The results show that compared with FORM and SORM, IS and MC-BPNN have higher calculation accuracy and efficiency and can more accurately evaluate the failure degree of composite structures and ensure their safe operation in the field of aerospace equipment. The universality of this method can make it being widely used in the reliability and sensitivity analysis of different composite materials as well as dispersion analysis.

WENO schemes using optimized third order fuzzy weight limiter functions

Abstract This work presents an improved version of non-linear weight limiters to obtain third order non-oscillatory WENO scheme. The construction of this modified weight limiter is based on fuzzy inference system, which is a knowledge based rule system. The linear combination of overlapped basis functions is used to achieve the optimized weight limiters by exploring the linguistics hedges operator on the basis functions. The WENO scheme using optimized weight limiter achieves third order of accuracy and gives higher resolution to discontinuities compared to other established third order WENO schemes.

An improved analytical solution to outcrossing rate for scalar nonstationary and non-gaussian processes

For time-dependent reliability methods, there is a lack of analytical solutions to outcrossing rates of nonstationary and non-Gaussian processes. This paper aims to propose an analytical method to determine this type of outcrossing rate with improved accuracy and efficiency. The novelty of this proposed method is that it can consider the probabilistic properties of non-Gaussian processes by combining a two-component parallel system model with higher-order moments-based reliability indexes. The main contributions of this method are: (1) compared with PHI2+ method, it is more accurate for the calculation of the outcrossing rate of nonstationary and non-Gaussian processes with nonlinear limit state functions; (2) compared with MPHI2 method, it is analytical and not only applicable for the nonstationary stochastic processes but also insensitive to time increments; and (3) it is not only more accurate than the existing methods for nonstationary and non-Gaussian processes but also more computation efficient than the Monte Carlo simulation method. The proposed method has shown its advantages for practical structures with neither Gaussian processes nor linear limit state functions, which are beneficial for both researchers and asset managers to evaluate the time-dependent reliability of structures accurately and to develop risk-informed maintenance schemes with a view to prolonging their service life.