Stochastic Input Parameters Research Articles

Trajectory-based global sensitivity analysis in multiscale models

This research introduces a novel global sensitivity analysis (GSA) framework for agent-based models (ABMs) that explicitly handles their distinctive features, such as multi-level structure and temporal dynamics. The framework uses Grassmannian diffusion maps to reduce output data dimensionality and sparse polynomial chaos expansion (PCE) to compute sensitivity indices for stochastic input parameters. To demonstrate the versatility of the proposed GSA method, we applied it to a non-linear system dynamics model and epidemiological and economic ABMs, depicting different dynamics. Unlike traditional GSA approaches, the proposed method enables a more general estimation of parametric sensitivities spanning from the micro level (individual agents) to the macro level (entire population). The new framework encourages the use of manifold-based techniques in uncertainty quantification, enhances understanding of complex spatio-temporal processes, and equips ABM practitioners with robust tools for detailed model analysis. This empowers them to make more informed decisions when developing, fine-tuning, and verifying models, thereby advancing the field and improving routine practice for GSA in ABMs.

Probabilistic machine learning approach to reliability analysis of a bogie frame under dynamic loading

ABSTRACT This study aims to evaluate the fatigue failure probability of a bogie frame considering the variability of input parameters, including loading ( L ), endurance limit ( S e ), and fillet size ( r ), through a data-driven surrogate model. Mechanical tests were conducted to determine the mechanical properties of the material of the bogie frame while a combination of machine learning and FEA has been utilized to generate a dataset for the dynamic response of the bogie frame under main in-service fatigue loads. Nine machine learning-based surrogate models were constructed based on the actual response at a limited set of data points chosen by the Optimum space-filling algorithm, and their accuracy was investigated. It is found that the CatBoost model is the optimal algorithm to map the stochastic input parameters with the factor of safety as the output parameter and perform the reliability evaluation. Also, results reveal a fatigue reliability of 99.34% for the bogie frame under normal conditions, with a cumulative failure probability of less than 0.66% over a 30-year service life. Furthermore, the results show that the proposed machine learning-based approach is an efficient tool to evaluate the fatigue failure probability of the bogie frame with reasonable accuracy when a small set of training data is available. This study’s scope extends to providing comprehensive guidelines for employing machine learning methods for fatigue reliability analysis of complex vehicle structures in the presence of various stochastic variables.

Modeling of Uncertainty Propagation for Transient Heat Rejection Problems

Generic plate and experiment-related high-fidelity models to study uncertainty propagation and transient processes for heat rejection problems are developed. The stochastic Biot numbers on the top and bottom surfaces of the plate, Bi1 and Bi2, and stochastic dimensionless input parameters, describing initial and convection boundary conditions, are introduced to define temperature variation in the modeled systems. The resulting uncertainty amplitude demonstrates a varying time evolution across a wide range of mean values of the input parameters. Depending on which input parameters are stochastic, the uncertainty may increase or decrease over time, or remain small for a given time interval in the case of the plate model. For small Bi1,2, dimensionless characteristic time for a temperature variation curve to approach its steady state is (Bi1+Bi2)−1 and large. An extremum in a temperature variation curve may appear when the Biot numbers are unequal and disappears if Bi1=Bi2. Results are nonsymmetric with respect to interchanging Bi1 and Bi2. Despite the difference in shape, the temperature variation profile over a specific region in the experimental test section numerically described by the high-fidelity model is close to the corresponding results provided by the plate model for the same Biot numbers, convection boundary, and initial conditions.

Experimental and numerical investigations on the stress waves propagation in strain-hardening fiber-reinforced cementitious composites: Stochastic analysis using polynomial chaos expansions

The propagation of stress waves can result in spalling fractures in materials, which poses a threat to the safety of engineering structures. Hence, in-depth studies are necessary to mitigate such risks. Ultra-high toughness cementitious composites (UHTCC) with significant tensile strain-hardening properties are advanced engineering materials with great potential for impact engineering. The study used a combination of experimental and numerical methods to analyse the propagation and attenuation behaviour of stress waves in the UHTCC. The numerical model was developed using the Smooth Particle Galerkin (SPG) method and the Karagozian & Case concrete/cementitious (KCC) material constitutive model. Additionally, a quantitative global sensitivity analysis (GSA) using a polynomial chaos expansion (PCE) surrogate model and Sobol's method was conducted to analyse the stochastic input parameters of the numerical model. Results show that compressive stress waves undergo significant attenuation during propagation in UHTCC, and the developed numerical model accurately simulated the stress wave propagation process Furthermore, the PCE-Sobol GSA revealed that the selected stress wave propagation responses are most sensitive to the tensile strength of UHTCC. This study is expected to deepen the understanding of the propagation behaviour of stress waves in UHTCC and pave the way for promising applications of UHTCC in impact engineering.

A multi-element non-intrusive Polynomial Chaos method using agglomerative clustering based on the derivatives to study irregular and discontinuous Quantities of Interest

A non-intrusive method to get a multi-element Polynomial Chaos model is developed. This method is called ME-ACD, for Multi-Element based on Agglomerative Clustering on Derivatives. It aims at approximating a Quantity of Interest which presents discontinuities or irregularities making it difficult to be accurately approximated by standard Polynomial Chaos models. The method permits to efficiently split the parameter space and to train local polynomial models of lower degrees on every element where the local pieces of the Quantity of Interest are smoother. The algorithm is based on agglomerative clustering of the observations in a well-chosen abstract space taking into account the value of the Quantity of Interest and of its derivatives with respect to the stochastic input parameters. The same observations are used for both partitioning the space and training the local models. Several partitions of the parameter space are tested, and the one leading to local models minimising a cross-validation error is selected. Once the training observations are labelled with a class number indicating the element they are located in, a neural network classifier is trained to determine which local model to use for further evaluations. The method has proven to efficiently split the parameter space for a set of applications of moderate dimension. The piecewise chaos model is compared with the ones of a standard Polynomial Chaos non-intrusive method and of a Gradient Boosted Trees method in terms of accuracy.

Polynomial neural network based probabilistic hydrodynamic analysis of two-lobe bearings with stochasticity in surface roughness

This paper investigates the probabilistic response of two-lobe bearings considering the uncertainty in eccentricity ratio, preload value, bearing clearance, supply pressure, oil viscosity and surface roughness. To simulate stochasticity in input variables, Monte Carlo simulation (MCS) is carried out in conjunction with the Reynolds equation using finite difference method. Polynomial neural network based machine learning model is used as a surrogate model to increase the efficiency of MCS. To assess the relative importance of the stochastic input parameters, a sensitivity analysis is carried out. The physically insightful new probabilistic results, presented here covering a wide spectrum of uncertainty sources including surface roughness, make it evident that different forms of source-uncertainties have a significant effect on the critical performance of bearings.

Vibration and Uncertainty Analysis of Functionally Graded Sandwich Plate Using Layerwise Theory

In the present study, free vibration analyses of porous functionally graded material (FGM) sandwich plates are carried out in thermal environment. Two types of sandwich plates such as sandwich with FGM face sheets and homogeneous core and sandwich with homogeneous face sheets and FGM core are considered for free vibration analysis with nonlinear temperature variation over the thickness direction. The metal and ceramic components are assumed to be temperature dependent. The higher-order layerwise theory is adopted with an eight-noded rectangular element to build a finite element model for free vibration analysis. A radial basis function network (RBFN) based surrogate model is developed to investigate the stochastic frequency response due to the uncertain material properties. The accuracy and the efficiency of the RBFN model are established by comparing the results with that of Monte Carlo simulation. Further, the developed surrogate model is implemented to perform various parametric studies and sensitivity analyses. The influence of the volume fraction index, types of porosity, and thermal gradient on the stochastic frequency response are analyzed. Though the influence of different stochastic input parameters is strongly dependent on the volume fraction index, the mass density of metal is found to be the most sensitive parameter for the sandwich plate with FGM face sheets and homogeneous core.

MANIFOLD LEARNING-BASED POLYNOMIAL CHAOS EXPANSIONS FOR HIGH-DIMENSIONAL SURROGATE MODELS

In this work we introduce a manifold learning-based method for uncertainty quantification (UQ) in systems describing complex spatiotemporal processes. Our first objective is to identify the embedding of a set of high-dimensional data representing quantities of interest of the computational or analytical model. For this purpose, we employ Grassmannian diffusion maps, a two-step nonlinear dimension reduction technique which allows us to reduce the dimensionality of the data and identify meaningful geometric descriptions in a parsimonious and inexpensive manner. Polynomial chaos expansion is then used to construct a mapping between the stochastic input parameters and the diffusion coordinates of the reduced space. An adaptive clustering technique is proposed to identify an optimal number of clusters of points in the latent space. The similarity of points allows us to construct a number of geometric harmonic emulators which are finally utilized as a set of inexpensive pre-trained models to perform an inverse map of realizations of latent features to the ambient space and thus perform accurate out-of-sample predictions. Thus, the proposed method acts as an encoder-decoder system which is able to automatically handle very high-dimensional data while simultaneously operating successfully in the small-data regime. The method is demonstrated on two benchmark problems and on a system of advection-diffusion-reaction equations which model a first-order chemical reaction between two species. In all test cases, the proposed method is able to achieve highly accurate approximations which ultimately lead to the significant acceleration of UQ tasks.

Uncertainty quantification in buckling strength of variable stiffness laminated composite plate under thermal loading

In this paper, thermal buckling analysis of variable stiffness composite laminate (VSCL) is carried out. The first-order shear deformation theory is adopted with eight noded isoparametric element to construct a finite element model for thermal buckling analysis. The effect of the uncertain composite properties that could arise due to complex manufacturing and fabrication process to the buckling temperature is investigated using polynomial neural network (PNN). The contribution of individual properties, such as the material properties, fiber orientations, and the thermal expansion coefficients of VSCL plate are performed for various boundary conditions and lamination sequences. A sensitivity analysis is also carried out and parameters to which buckling temperature has high sensitivity are identified. The accuracy and efficiency of PNN model are compared with Monte Carlo simulation (MCS). Further, PNN is employed to evaluate the reliability parameters of VSCL plate. From the various studies, it is observed that even though the influence of different stochastic input parameters are depend on the boundary conditions, the transverse coefficient of thermal expansion (α22) is found to be most influential parameter, followed by transverse (E22) and longitudinal (E11) Young’s modulus, respectively.

Real-time inverse solution of the composites’ cure heat transfer problem under uncertainty

This paper addresses the development of an inversion scheme based on Markov Chain Monte Carlo integrating process modelling with monitoring data for the real-time probabilistic estimation of unknown stochastic input parameters such as heat transfer coefficient and resin thermal conductivity and process outcomes during the manufacture of fibrous composites materials. Kriging was utilized to build an efficient surrogate model of the composite curing process based on finite element modelling. The utilization of an inverse scheme with real-time temperature monitoring driving the estimation of process parameters during manufacture results in real-time probabilistic prediction of process outcomes.

Assessing the impact of uncertainty and the level of aggregation in case mix planning

In this article, we present a framework for evaluating the impact of uncertainty and the use of different aggregation levels in case mix planning on the quality of strategic decisions regarding the case mix of a hospital. In particular, we analyze the effect of modeling (i) demand, (ii) resource use, and (iii) resource availability as stochastic input parameters on the performance of case mix planning models. In addition, the consequences of taking the weekly structure with inactive days without surgeries into account are assessed (iv). The purpose of this paper is to provide a guideline for the decision-maker planning the case mix on the consideration of stochastic aspects and different aggregation levels. We formulate a mixed integer programming model for case mix planning along with different stochastic and deterministic extensions. The value of the different extensions is analyzed using a factorial design. The resulting stochastic models are solved using sample average approximation. Simulation is used to evaluate the strategies derived by the different models using real-world data from a large German hospital. We find that highly aggregated basic case mix planning models can overestimate the objective value by up to 10% and potentially lead to biased results. Refining the problem decreased the gap between projected case mix planning results and simulated results considerably and led to improved solutions.

Spatially varying fuzzy multi-scale uncertainty propagation in unidirectional fibre reinforced composites

This article presents a non-probabilistic fuzzy based multi-scale uncertainty propagation framework for studying the dynamic and stability characteristics of composite laminates with spatially varying system properties. Most of the studies concerning the uncertainty quantification of composites rely on probabilistic analyses, where the prerequisite is to have the statistical distribution of stochastic input parameters. In many engineering problems, these statistical distributions remain unavailable due to the restriction of performing large number of experiments. In such situations, a fuzzy-based approach could be appropriate to characterize the effect of uncertainty. A novel concept of fuzzy representative volume element (FRVE) is developed here for accounting the spatially varying non-probabilistic source-uncertainties at the input level. Such approach of uncertainty modelling is physically more relevant than the prevalent way of modelling non-probabilistic uncertainty without considering the ply-level spatial variability. An efficient radial basis function based stochastic algorithm coupled with the fuzzy finite element model of composites is developed for the multi-scale uncertainty propagation involving multi-synchronous triggering parameters. The concept of a fuzzy factor of safety (FFoS) is discussed in this paper for evaluation of safety factor in the non-probabilistic regime. The results reveal that the present physically relevant approach of modelling fuzzy uncertainty considering ply-level spatial variability obtains significantly lower fuzzy bounds of the global responses compared to the conventional approach of non-probabilistic modelling neglecting the spatially varying attributes.

Stochastic low-velocity impact on functionally graded plates: Probabilistic and non-probabilistic uncertainty quantification

This paper quantifies the compound effect of source-uncertainties on low-velocity impact of functionally graded material (FGM) plates following a coupled surrogate based finite element simulation approach. The power law is employed to evaluate the material properties of FGM plate at different points, while the modified Hertzian contact law is implemented to determine the contact force and other parameters in a stochastic framework. The time dependent equations are solved by Newmark's time integration scheme. Insightful results are presented by investigating the effects of degree of stochasticity, oblique impact angle, thickness of plate, temperature, power law index, and initial velocity of impactor following both probabilistic and non-probabilistic approaches along with in-depth deterministic analyses. A detail probabilistic analysis leading to complete probabilistic characterization of the structural responses can be carried out when the statistical distribution of the stochastic input parameters are available. However, in many cases concerning FGM, these statistical distributions may remain unavailable due to the restriction of performing large number of experiments. In such situations, a fuzzy-based non-probabilistic approach could be appropriate to characterize the effect of uncertainty. A surrogate based approach based on artificial neural network coupled with the finite element model for low-velocity impact analysis of FGM plates is developed for achieving computational efficiency. The numerical results reveal that the low-velocity impact on FGM plates is significantly influenced by the effect of inevitable source-uncertainty associated with the stochastic system parameters, whereby the importance of adopting an inclusive design paradigm considering the effect of source-uncertainties in the impact analysis is established.

On the propagation of uncertainties in radiation belt simulations

We present the first study of the uncertainties associated with radiation belt simulations, performed in the standard quasi-linear diffusion framework. In particular, we estimate how uncertainties of some input parameters propagate through the nonlinear simulation, producing a distribution of outputs that can be quite broad. Here, we restrict our focus on two-dimensional simulations (in energy and pitch angle space) of parallel-propagating chorus waves only, and we study as stochastic input parameters the geomagnetic index Kp(that characterizes the time dependency of an idealized storm), the latitudinal extent of waves, and the average electron density. We employ a collocation method, thus performing an ensemble of simulations. The results of this work point to the necessity of shifting to a probabilistic interpretation of radiation belt simulation results, and suggest that an accurate specification of a time-dependent density model is crucial for modeling the radiation environment.

Stochastic natural frequency analysis of damaged thin-walled laminated composite beams with uncertainty in micromechanical properties

This paper presents a stochastic approach to study the natural frequencies of thin-walled laminated composite beams with spatially varying matrix cracking damage in a multi-scale framework. A novel concept of stochastic representative volume element (SRVE) is introduced for this purpose. An efficient radial basis function (RBF) based uncertainty quantification algorithm is developed to quantify the probabilistic variability in free vibration responses of the structure due to spatially random stochasticity in the micro-mechanical and geometric properties. The convergence of the proposed algorithm for stochastic natural frequency analysis of damaged thin-walled composite beam is verified and validated with original finite element method (FEM) along with traditional Monte Carlo simulation (MCS). Sensitivity analysis is carried out to ascertain the relative influence of different stochastic input parameters on the natural frequencies. Subsequently the influence of noise is investigated on radial basis function based uncertainty quantification algorithm to account for the inevitable variability for practical field applications. The study reveals that stochasticity/ system irregularity in structural and material attributes affects the system performance significantly. To ensure robustness, safety and sustainability of the structure, it is very crucial to consider such forms of uncertainties during the analysis.

On the quasi-Monte Carlo method with Halton points for elliptic PDEs with log-normal diffusion

This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings. This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings.

Uncertainty Quantification in Natural Frequency of Composite Plates - An Artificial Neural Network Based Approach

This paper presents the stochastic natural frequency for laminated composite plates by using artificial neural network (ANN) model. The ANN model is employed as a surrogate and is trained by using Latin hypercube sampling. Subsequently the stochastic first two natural frequencies are quantified with ANN based uncertainty quantification algorithm. The convergence of the proposed algorithm for stochastic natural frequency analysis of composite plates is verified and validated with original finite element method (FEM) in conjunction with Monte Carlo simulation. Both individual and combined variation of stochastic input parameters are considered to address the influence on the output of interest. The sample size and computational cost are reduced by employing the present approach compared to traditional Monte Carlo simulation.

Uncertain natural frequency analysis of composite plates including effect of noise – A polynomial neural network approach

This paper presents the quantification of uncertain natural frequency for laminated composite plates by using a novel surrogate model. A group method of data handling in conjunction to polynomial neural network (PNN) is employed as surrogate for numerical model and is trained by using Latin hypercube sampling. Subsequently the effect of noise on a PNN based uncertainty quantification algorithm is explored in this study. The convergence of the proposed algorithm for stochastic natural frequency analysis of composite plates is verified and validated with original finite element method (FEM). Both individual and combined variation of stochastic input parameters are considered to address the influence on the output of interest. The sample size and computational cost are reduced by employing the present approach compared to direct Monte Carlo simulation (MCS).

Reliability analysis of static liquefaction of loose sand using the random finite element method

Purpose– Liquefaction of soils is defined as significant reduction in shear strength and stiffness due to increase in pore water pressure. This phenomenon can occur in static (monotonic) or dynamic loading patterns. However, in each pattern, the inherent variability of the soil parameters indicates that this problem is of a probabilistic nature rather than being deterministic. The purpose of this paper is to present a method, based on random finite element method, for reliability assessment of static liquefaction of saturated loose sand under monotonic loading.Design/methodology/approach– The random finite element analysis is used for reliability assessment of static liquefaction of saturated loose sand under monotonic loading. The soil behavior is modeled by an elasto-plastic effective stress constitutive model. Independent soil parameters including saturated unit weight, peak friction angle and initial plastic shear modulus are selected as stochastic parameters which are modeled using a truncated normal probability density function (pdf).Findings– The probability of liquefaction is assessed by pdf of modified pore pressure ratio at each depth. For this purpose pore pressure ratio is modified for monotonic loading of soil. It is shown that the saturated unit weight is the most effective parameter, within the selected stochastic parameters, influencing the static soil liquefaction.Originality/value– This research focuses on the reliability analysis of static liquefaction potential of sandy soils. Three independent soil parameters including saturated unit weight, peak friction angle and initial plastic shear modulus are considered as stochastic input parameters. A computer model, coded in MATLAB, is developed for the random finite element analysis. For modeling of the soil behavior, a specific elasto-plastic effective stress constitutive model (UBCSAND) was used.

Modeling of dynamic fracture based on the cracking particles method

The dynamic fragmentation of brittle materials using the cracking particles method (CPM) with obscuration zone is studied. The CPM is an effective meshfree method for arbitrary evolving cracks. The crack is modeled by piecewise straight crack segments and does not require any topological representation of the crack surface. To avoid artificial cracks observed in discrete continuum approaches, obscuration zones are used as suggested by Mott. The influence of the variation in material properties with different stochastic input parameters on the results is studied as well.