Probabilistic Engineering Mechanics最新文献

{"title":"Efficient Wiener path integral analysis of high-dimensional nonlinear systems via reduced-order representation and path extrapolation","authors":"Ilias G. Mavromatis,&nbsp;Ketson R.M. dos Santos,&nbsp;Ioannis A. Kougioumtzoglou","doi":"10.1016/j.probengmech.2026.104042","DOIUrl":"10.1016/j.probengmech.2026.104042","url":null,"abstract":"<div><div>A novel computational methodology is developed for enhancing the efficiency of the Wiener path integral (WPI) technique in determining the stochastic response of high-dimensional nonlinear dynamical systems. Specifically, within the WPI technique, the system response probability density function (PDF) is represented as a functional integral over the space of admissible response trajectories. In general, this functional integral is evaluated approximately by retaining its dominant contribution associated with the most probable path. This is determined through a functional minimization problem equivalent to the solution of a deterministic boundary value problem (BVP). However, in a brute-force implementation, evaluation of the response PDF requires discretization of the PDF domain and the numerical solution of one BVP for each grid point; thus rendering the technique computationally prohibitive for high-dimensional systems. To address this computational challenge, a reduced-order WPI formulation is developed for the efficient determination of lower-dimensional joint PDFs corresponding to selected response variables of interest, without requiring explicit resolution of the full stochastic state space. Specifically, the developed methodology combines an efficient Rayleigh–Ritz representation of the most probable path problem with mixed fixed/free boundary conditions, thereby recasting the original functional minimization problem into a finite-dimensional optimization problem. Next, a novel extrapolation strategy is introduced directly in the reduced coefficient space, whereby most probable paths corresponding to neighboring points in the reduced PDF domain are constructed through a decomposition into deterministic boundary-induced corrections and coefficient-space updates. Further, by invoking a local linearization of the stochastic action functional, the transition between neighboring points in the PDF domain is reduced to the solution of a low-dimensional quadratic optimization problem whose governing matrices can be precomputed independently of the grid discretizing the PDF domain. Consequently, the computational complexity of the WPI technique is drastically reduced by orders of magnitude compared to previous formulations. A representative high-dimensional nonlinear multiscale nanorod system involving coupled coarse–fine dynamics and stochastic excitation acting across multiple scales is considered to demonstrate the capabilities of the proposed framework. Comparisons with Monte Carlo simulation data demonstrate excellent agreement while requiring only a limited number of BVP solutions, thereby validating both the accuracy and computational efficiency of the developed methodology.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104042"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149155310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A probabilistic bidirectional nonlinear dynamic framework for vibratory compactor-soil interaction with mechanical inertance and hybrid contact","authors":"Dong Feng","doi":"10.1016/j.probengmech.2026.104061","DOIUrl":"10.1016/j.probengmech.2026.104061","url":null,"abstract":"<div><div>Vibratory compaction is widely used in road engineering to improve the mechanical stability and service performance of compacted soil layers, while its effectiveness depends strongly on the coupled dynamic response of the machine and soil. Existing compactor-soil models have clarified important vertical vibration mechanisms, but many formulations remain deterministic, unidirectional, or insufficiently equipped to describe mechanical inertance, intermittent contact, frictional interaction, and uncertainty-driven failure events within a unified framework. This study develops a probabilistic bidirectional nonlinear dynamic model incorporating mechanical inertance, nonlinear frame-drum suspension, hybrid loading, unloading, and separation behavior, tangential frictional contact, and work-controlled compaction evolution. Correlated colored stochastic processes are introduced to represent uncertain excitation and soil properties while preserving physically admissible parameter values. The resulting stochastic hybrid system is evaluated through first-passage separation, sliding, and compaction events, together with reliability, fragility, upper-tail response, spectral, recurrence, and stochastic-stability diagnostics. The framework provides a unified route for linking nonlinear dynamic response, uncertainty propagation, and performance-oriented risk measures in vibratory compaction.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104061"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149252946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Time variable step integrator for evolutionary Lyapunov covariance equation in non-stationary random vibration","authors":"Giuseppe Carlo Marano","doi":"10.1016/j.probengmech.2026.104048","DOIUrl":"10.1016/j.probengmech.2026.104048","url":null,"abstract":"<div><div>The second-order response statistics of linear structures driven by non-stationary Gaussian excitations are governed by the evolutionary (differential) Lyapunov matrix equation. In random-vibration and seismic reliability practice this equation is integrated almost exclusively with fixed-step schemes that carry no local error control and no explicit stability safeguard, so that accuracy and stability hinge on a manually chosen time step. We propose a variable-step integration framework that removes this dependence while preserving the matrix structure exploited by the existing fixed-step schemes: each (sub)step still reduces to a single algebraic Lyapunov solve. Two members are developed: an adaptive implicit trapezoidal scheme with a step-doubling (Richardson) local error estimate, and an L-stable TR-BDF2 scheme with an embedded local error estimate requiring two Lyapunov solves per step. Both are driven by a PI step-size controller and are equipped with an A-stability-aware diagnostic. On shear frames with classical and non-classical damping, with up to 500 degrees of freedom, excited by an amplitude- and frequency-modulated Clough–Penzien process, the methods are validated against an independent high-order reference and compared, through work–precision diagrams, with the ready-to-use baseline obtained by half-vectorising the equation and calling a stiff ODE solver, and with an explicit Runge–Kutta solver. When the filter frequency is modulated in time, so that the state matrix becomes time dependent, the proposed structure-preserving integrators are several times faster than the half-vectorised stiff baseline and the gap widens with the number of degrees of freedom, because the baseline must refactorise the Jacobian of a lifted system of dimension <span><math><mrow><mi>m</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></math></span> at every step—and becomes infeasible beyond a few tens of degrees of freedom for lack of memory. Against explicit integration the outcome is governed by the stiffness of the system: for soft frames on soft soil the explicit solver remains cheaper and the proposed schemes offer robustness rather than speed, whereas beyond a highest frequency of about 50<!--> <!-->rad/s (stiff low-rise buildings, secondary systems, rock-site input) the L-stable scheme is faster by more than an order of magnitude. The framework yields the covariance history directly, in a form immediately usable for first-passage reliability.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104048"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149253026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Probabilistic reliability modeling of delayed degradation systems with mixed random shocks and discontinuous state jumps","authors":"Hao Lyu,&nbsp;Jing Pan,&nbsp;Chuanhai Chen,&nbsp;Murat Ozkut,&nbsp;Michael Pecht","doi":"10.1016/j.probengmech.2026.104047","DOIUrl":"10.1016/j.probengmech.2026.104047","url":null,"abstract":"<div><div>Many engineering components experience a degradation delay period before observable deterioration becomes dominant. In random shock environments, this delay period may be shortened by external shocks, and the subsequent degradation process can involve discontinuous jumps among damage states. To characterize these coupled effects, this paper develops a two-stage probabilistic reliability model for delayed degradation systems subject to mixed random shocks. The first stage describes the degradation delay period, whose termination can be accelerated by shock-induced damage accumulation or run-shock events. The second stage is formulated as a multi-state degradation process in which shocks may induce non-adjacent state jumps. Based on a mixed-shock mechanism and a finite Markov chain embedding framework, the hard-failure process is analytically modeled. Soft failure is described by a path-dependent linear degradation model with state-dependent degradation rates and shock-induced damage increments. The overall reliability is derived using the law of total probability by integrating the transition-time distribution and conditional survival probabilities in the degradation stage. A case study illustrates the framework and shows that ignoring degradation delay or shock-induced state jumps may lead to biased reliability assessment.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104047"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149155307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Statistical scaling of safety factors for buckling of spherical shells with non-interacting defects","authors":"Yangyi Liu,&nbsp;Roberto Ballarini,&nbsp;Jia-Liang Le","doi":"10.1016/j.probengmech.2026.104046","DOIUrl":"10.1016/j.probengmech.2026.104046","url":null,"abstract":"<div><div>The resistance to buckling of a thin-walled structure subjected to a deterministic load exhibits significant variability due to the random nature of inevitable imperfections in its nominal geometry. The risk of buckling is increased if the loading is also random. To account for these uncertainties, the design of such structures requires the use of appropriate safety factors. This study presents a mechanistic model for the safety factors of pressurized spherical shells containing non-interacting Gaussian-shaped localized defects with a fixed normalized width but random amplitudes, and subjected to uniform pressure of random magnitude. The probability distribution of the buckling pressure is modeled using a finite weakest-link approach. The results show that for a given load distribution the failure probability increases with increasing number of defects (the maximum number of non-interacting defects being proportional to the shell’s dimensionless radius) and approaches a constant value. The model is used to determine the central safety factor, defined as the ratio of the mean buckling pressure to the mean applied load, for a specified target failure probability. The safety factor is shown to strongly depend on the dimensionless radius, a result which is understood as a form of statistical scaling. This finding suggests that the current practice of using a constant and size-independent safety factor cannot ensure a uniform target failure probability across all shell sizes.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104046"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149155308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Prescribed time adaptive fault-tolerant control of direct-drive wave energy converter system","authors":"Shizhan Dong,&nbsp;Zhongqiang Wu,&nbsp;Zijian Zhang","doi":"10.1016/j.probengmech.2026.104039","DOIUrl":"10.1016/j.probengmech.2026.104039","url":null,"abstract":"<div><div>A prescribed time adaptive fault-tolerant controller is designed for the problem of maximum power point tracking in <span><span>Direct-Drive Wave Energy Converter</span><svg><path></path></svg></span>System (DWECS), considering actuator fault, internal parameters change and external disturbances. First, the model of DWECS with external disturbances is established, and the actuator fault, system parameters change and external disturbances are regarded as lumped uncertainties. An adaptive fault-tolerant controller based on backstepping method with prescribed time theory is designed to meet the requirements of rapidity and stability for maximum power point tracking control. By designing the prescribed time observer, the lumped uncertainties can be estimated accurately in the prescribed time, and introduced into the controller to compensate for them. The prescribed time adaptive fault-tolerant controller does not depend on the initial state of the system and the parameters of controller, and the system can still maintain the maximum power point tracking and achieve the prescribed time stability.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104039"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149155306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fractional calculus-based contact mechanics in stochastic rocking dynamics of rigid blocks","authors":"Yuanjin Zhang,&nbsp;Ioannis P. Mitseas,&nbsp;Vasileios C. Fragkoulis","doi":"10.1016/j.probengmech.2026.104051","DOIUrl":"10.1016/j.probengmech.2026.104051","url":null,"abstract":"<div><div>A semi-analytical fractional-order stochastic linearization and averaging framework is developed for assessing the toppling probability of rigid rocking blocks on flexible foundations subjected to evolutionary ground motion excitation. The proposed formulation extends classical calculus-based foundation-contact rocking models by introducing a generalized fractional Winkler-type foundation-contact law, thereby embedding hereditary foundation-contact interaction effects. These memory effects provide a phenomenological means of representing complex soil-block interfaces, such as saturated or partially saturated contact conditions, where delayed drainage and capillary mechanisms may induce history-dependent behavior. The assembled rocking dynamics are nonlinear due to unilateral contact, uplift, geometry-dependent contact kinematics, and the response-dependent contact region. By combining static condensation with a stochastic linearization and averaging approach tailored to unbounded response behavior, the proposed method captures the threshold-crossing characteristics associated with toppling while retaining the specific mechanics of rocking uplift and foundation-contact interaction. The non-stationary probability density function of the rocking-amplitude is formulated to efficiently quantify the time-evolving toppling probability. The technique offers significant computational advantages over direct numerical simulation while accounting for the effects of time-dependent excitation intensity and frequency content. Numerical examples involving different block configurations, foundation models, and seismic intensity levels demonstrate the influence of fractional memory, foundation flexibility, uplift, and excitation intensity on the evolution of toppling probability. Comparisons with Monte Carlo simulation (MCS) data confirm the accuracy and reliability of the method, highlighting its potential for probabilistic assessment of rocking systems in seismic engineering applications.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104051"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149205039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Physics-constrained prediction of first-passage-time distributions in a strongly nonlinear stochastic Φ6 oscillator","authors":"Hongtao Chu,&nbsp;Changshui Feng,&nbsp;Juyong Zhang","doi":"10.1016/j.probengmech.2026.104062","DOIUrl":"10.1016/j.probengmech.2026.104062","url":null,"abstract":"<div><div>Accurate reconstruction of first-passage-time (FPT) distributions is essential for time-dependent reliability analysis of nonlinear stochastic systems but is costly when repeated large-scale Monte Carlo simulations are required. This study proposes a physics-constrained hybrid surrogate for a stochastic <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mn>6</mn></mrow></msup></math></span> oscillator. An energy-based three-parameter form represents the FPT density. Target-specific, dimensionally consistent features are derived using the Buckingham <span><math><mi>π</mi></math></span> theorem. Ridge regression predicts the three density parameters, and separate Gaussian process models correct their residuals. The model is evaluated on fixed in-distribution (ID) and directional out-of-distribution (OOD) test sets generated by directional extrapolation. For the 69 samples yielding admissible predictions from both models, the hybrid model increases the median PDF-level <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> from 0.9185 to 0.9633. It reduces the median Kolmogorov–Smirnov statistic, AD-type discrepancy, and Wasserstein <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> distance from 0.0719 to 0.0463, from 1.9625 to 0.4224, and from 3.6966 s to 1.6214 s, respectively. The ridge baseline and hybrid model yield admissible parameter triplets for 81 and 82 of 100 test samples, respectively. These results demonstrate improved FPT distribution reconstruction on the common admissible subset, while the remaining inadmissible predictions indicate that parameter admissibility can be further improved.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104062"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149282903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A deep-learning enhanced probability flow tube method for probability density evolution analysis of nonlinear structures","authors":"Futai Zhang,&nbsp;Zheyi Guo,&nbsp;Jun Xu","doi":"10.1016/j.probengmech.2026.104049","DOIUrl":"10.1016/j.probengmech.2026.104049","url":null,"abstract":"<div><div>Probability density evolution analysis of nonlinear structures provides a basis for uncertainty propagation and dynamic reliability assessment. The recently developed Probability Flow Tube Method (PFTM) enables the reconstruction of the time-dependent Probability Density Function (PDF) of a quantity of interest without numerically solving the generalized density evolution equation. However, applying PFTM to complex nonlinear structures may require a large ensemble of deterministic response time histories, resulting in considerable computational cost. To address this issue, this study proposes a deep-learning enhanced PFTM framework. A hybrid temporal network is trained on a small number of excitation–response pairs and used to predict a large ensemble of response trajectories for subsequent PDF reconstruction. The predicted trajectories are organized into time-continuous probability flow tubes with time-invariant probability weights. The evolving mean and standard deviation of each tube are determined algebraically by matching characteristic-function Taylor coefficients at the origin, and the global PDF is reconstructed as a probability-weighted mixture of the tube PDFs. The framework is evaluated using a real-world engineering example involving stochastic seismic response analysis of an onshore 5-MW steel–concrete hybrid wind turbine tower. The reconstructed PDF evolution and tail probabilities show close agreement with the corresponding Monte Carlo simulation results obtained using <span><math><mrow><mn>1</mn><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span> samples. These results indicate that the deep-learning model can efficiently generate the response ensemble required by PFTM while substantially reducing the number of computationally demanding time–history analyses of nonlinear structures.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104049"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149200963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multi-level error propagation of distance models in Gaussian random field generation over manifolds","authors":"Yan-Ping Liang,&nbsp;Shixue Liang,&nbsp;Miaomiao Sun,&nbsp;Xiaodan Ren","doi":"10.1016/j.probengmech.2026.104055","DOIUrl":"10.1016/j.probengmech.2026.104055","url":null,"abstract":"<div><div>This study examines the practical consequence of distance model selection in Gaussian random field generation over manifolds. An exponential correlation model is considered on three representative surfaces: a half-cylinder, a hemisphere, and a saddle surface. For each manifold, three distance models are compared: the reference geodesic distance computed on the triangulated surface using the MMP algorithm, the approximate geodesic distance obtained by the heat method, and the Euclidean distance computed from Cartesian coordinates. Errors are evaluated successively at the distance matrix, covariance matrix, and random field levels. The results show that higher distance-matrix accuracy does not necessarily lead to lower field-level discrepancy. Although the heat-method distance is consistently closer to the reference geodesic distance than the Euclidean distance, this geometric advantage is not preserved after covariance construction and field generation. In all investigated cases, the Euclidean distance model produces smaller field-level discrepancies and requires the lowest computational cost. These results suggest that distance model selection for manifold based random field simulation should be based on the propagated effect on the covariance structure and generated random fields, rather than on geometric distance accuracy alone.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104055"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149205035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}