Stochastic Finite Element Method Research Articles

Uncertainty quantification in the modal analysis of aircraft stiffeners: A Perturbation Technique approach in SFEM

Structural members, such as stiffeners, are crucial in aeronautical structures, providing essential dynamic stiffness. However, variability introduced by manufacturing and assembly processes, as well as material inconsistencies, can lead to uncertainties in structural performance. It is crucial to incorporate these uncertainties into structural analysis to ensure reliable design. This paper utilizes the Stochastic Finite Element Method (SFEM) to address uncertainties in typical structural members used in aeronautics. The Perturbation Technique, based on Taylor series expansions, was employed to model uncertainties in aircraft stiffeners. The study focuses on natural frequencies and modal analysis to evaluate the impact of uncertainties on beams with hat and Z sections, commonly used as stiffeners in aircraft panels. These stiffeners were modeled using the Timoshenko beam theory, and sensitivity analysis was performed to identify key contributors to variability. The perturbation parameter was validated through Monte Carlo simulations. Sensitivity analysis, employing gradient-based methods, identified significant factors affecting variability in natural frequencies. A different perturbation parameter was necessary based on the stiffener's geometry: thickness variations required a perturbation parameter on the order of 10−3, whereas dimensions changes in the flange and height required parameters on the order of 10−2. These results underscore the importance of choosing appropriate perturbation magnitudes to avoid inaccuracies in the deterministic frequency response. Once a perturbation parameter is established, it can be applied to similar regions, ensuring the robustness of the SFEM methodology in analyzing the dynamic response of aeronautical structural reinforcements.

Quantitative analysis of construction risks in shallowly buried biased tunnel portal section

To address the prominent problem of collapse instability in shallow buried soft ground tunnels, a non-invasive stochastic finite element method was introduced. Taking Fujian Puyan Wenbishan tunnel as the background project, ABAQUS finite element software was used to analyze the tunnel excavation mechanics and parameter sensitivity. And developed the software interface program based on Python to output explicit limit state equation for the key mechanical indexes of the tunnel, so as to evaluate the tunnel reliability under different excavation methods, quantitatively. Study results show a significant improvement in efficiency and accuracy when calculating the probability of failure in tunnel excavation by the non-invasive stochastic finite element method. The maximum displacement monitoring points for the Wenbishan tunnel portal section were all vault settlement, with displacements of 33.6 mm, 30.2 mm, and 25.3 mm, respectively, using the annular retained core soil method, single sidewall guide pit and double sidewall guide pit method, with probabilities of failure of 36.11%, 28.03%, and 20.02%. It is found that the reliability of the tunnel is mainly determined by the geotechnical weight, elastic modulus and cohesion of the weak sandy soil layer, which can provide ideas for this type of engineering researches.

On application of the relative entropy concept in reliability assessment of some engineering cable structures

The main research problem studied in this work is an uncertain response and reliability assessment of the spatial cable structures due to the environmental stochasticity as well as material and geometrical imperfections. Some popular cable structures are analyzed for this purpose using the Stochastic Finite Element Method (SFEM) implemented with the use of three different techniques, namely the iterative generalized perturbation method, semi-analytical approach as well as the Monte-Carlo simulation. Uncertainty quantification delivered in this study is based on the series of FEM analyses of both static and dynamic structural problems. They enable the Least Squares Method determination of the structural polynomial responses linking extreme stresses and deformations with several uncorrelated uncertainty sources. Reliability assessment, fundamental in durability and Structural Health Monitoring, is completed using a comparison of the First Order Reliability Method (FORM) with probabilistic distance formulated by Bhattacharyya. Input uncertainties are assumed to be Gaussian according to the Maximum Entropy Principle. They have specific expected values following engineering design demands or the provisions of designing codes, whereas their standard deviations do not exceed the 10% level. The methods presented and the results obtained in this study may serve for further reliability analyses of large-scale civil engineering structures completed with both steel cables and also reinforced concrete plates like suspended bridges, for instance.

Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace

The stochastic finite element method is an important tool for structural reliability analysis. In order to improve the calculation efficiency, a stochastic finite element method based on the Krylov subspace is proposed for the static reliability analysis of structures. The first step of the proposed method is to preprocess the static response equation considering randomness to reduce the condition number of the coefficient matrix. The second step of the proposed method is to construct a Krylov subspace based on the preprocessed static response equation. Then, the static displacement of random sampling is expressed as a linear combination of subspace basis vectors to achieve the purpose of a fast solution. Finally, statistics and failure probability are calculated according to the static response obtained from thousands of random samples. Three numerical examples are given to compare the proposed method with the stochastic finite element method based on the Neumann series. The results show that the stochastic finite element method based on the Krylov subspace is more accurate and efficient than the stochastic finite element method based on the Neumann series.

Uncertainty quantification for natural frequency and mode of variable angle tow composite plates with random and interval uncertainties

Variable angle tow (VAT) composite laminated structures are produced by automated tow-placement technology which can take advantage of the designability of composites to meet the aerospace industry’s need for dynamic properties. Complex production techniques inevitably lead to uncertainties in material properties due to manufacturing defects. The aim of this work is to present an efficient computational method for estimating the dynamical properties of VAT composite laminated plates due to multiscale hybrid uncertainties. This method consists of perturbation stochastic finite element method and particle swarm optimization combined with Mori-Tanaka homogenization. The degree of influence of the multiscale parameters on the dynamic properties of VAT composites with different structural forms has been obtained by sensitivity analyses of the frequency and mode. The method is applied to investigate variations in natural frequency and mode of a variety of VAT composite laminated plates with different fiber tow paths, layers, and boundary conditions. The results indicate that variations in natural frequencies and mode shapes are strongly linked to fiber tow paths and boundary conditions. The frequency and modes of the VAT show a nonlinear variation with angle. This method can quickly obtain the probability distribution intervals of the frequency, which is of great significance for assessing the reliability of VAT composites laminated plates dynamic designs.

Seismic reliability analysis of high earth-rockfill dams subjected to mainshock-aftershock sequences using a novel noninvasive stochastic finite element method

To eliminate the limitations of the deterministic analysis approach currently used for seismic safety evaluation of dams, this paper introduces a novel framework for analyzing the safety of high dams based on probability and reliability. This framework incorporates the randomness nature and impact of aftershocks in simulating main-aftershock sequences. Then, an improved generalized plastic model (GPM) is adopted to perform static-dynamic deformation analysis of the dam, which is achieved by only one set of parameters. Lastly, the non-invasive stochastic finite element method (SFEM), combined with the direct probability integral method (DPIM), is adopted. This method yields probability density functions (PDFs) for each seismic response moment, ensuring both efficient and accurate assessment of dynamic time-varying reliability. The new framework was employed to evaluate the deformation reliability and slope stability of Lianghekou dam. Results demonstrate its superior efficiency compared to conventional methods, verifying its suitability for stochastic dynamic analysis of large and complex nonlinear structures. The study suggests that the variable impact of aftershocks on dam stability depends on design requirements, emphasizing the necessity to consider such effects in dam safety assessments.

Methods for characterization and continuum modeling of inhomogeneous properties of paper and paperboard materials: A review

The potential of paper and paperboard as fiber-based materials capable of replacing conventional polymer-based materials has been widely investigated and evaluated. Due to paper’s limited extensibility and inherent heterogeneity, local structural variations lead to unpredictable local mechanical behavior and instability during processing, such as mechanical forming. To gain a deeper understanding of the impact of mechanical behavior and heterogeneity on the paper forming process, the Finite Element Method (FEM) coupled with continuum modeling is being explored as a potential approach to enhance comprehension. To achieve this goal, utilizing experimentally derived material parameters alongside stochastic finite element methods allows for more precise modeling of material behavior, considering the local material properties. This work first introduces the approach of modeling heterogeneity or local material structure within continuum models, such as the Stochastic Finite Element Method (SFEM). A fundamental challenge lies in accurately measuring these local material properties. Experimental investigations are being conducted to numerically simulate mechanical behavior. An overview is provided of experimental methods for material characterization, as found in literature, with a specific focus on measuring local mechanical material structure. By doing so, it enables the characterization of the global material structure and mechanical behavior of paper and paperboard.

An efficient Bayesian method with intrusive homotopy surrogate model for stochastic model updating

AbstractThis paper proposes a new stochastic model updating method based on the homotopy surrogate model (HSM) and Bayesian sampling. As a novel intrusive surrogate model, the HSM is established by the homotopy stochastic finite element (FE) method. Then combining the advanced delayed‐rejection adaptive Metropolis–Hastings sampling technology with HSM, the structural FE model can be updated by uncertain measurement modal data. The numerical results show that the updating effectiveness of the proposed method is better than that of the Bayesian methods with the non‐intrusive surrogate models, such as stochastic response surface model and Kriging model. Compared to the Bayesian method with the intrusive second‐order perturbation model, the updating results of the proposed method are more accurate, especially when the fluctuation of the uncertain measured data is large and the stiffness of the structure significantly changes. The model updating results of a cable‐stayed bridge show that the statistic modal properties of the updated bridge model have a very good agreement with the uncertain measurement modal data.

Reciprocal cross‐correlation analysis of two‐phase seepage processes and reservoir heterogeneities in CO2 saline aquifer sequestration

AbstractWhen CO2 saline aquifer storage is carried out, the heterogeneity of reservoir rock is an important factor affecting CO2 transport, and the reservoir heterogeneity in numerical simulations is mainly manifested as the heterogeneity of the parameter field. Since the parameter distributions across the reservoir are not available with the existing probes, the stochastic finite element method is combined with a two‐phase flow model to establish an unconditional random field of permeability, and computations are performed using the Monte Carlo method. The permeability, CO2 maximum migration distance (Md) and CO2 sweep area (Sa) were analyzed for mutual correlation. The permeability correlation area affecting Md and Sa was obtained, and the changes in the correlation area under the coefficient of variation (Cv) and correlation length (λx) of the permeability field in the different reservoirs were analyzed. The kriging superposition approach (KSA) was subsequently used to estimate both the Md and Sa of the target reservoir by establishing conditional random fields based on the sampling parameters in regions with different correlations, resulting in errors of 0.66% for Md and 0.96% for Sa in the high correlation region and 4.86% and 3.12% for Md and Sa in the low correlation region, which suggested that the sampling results from the high correlation region were less biased in the estimation. Under limited sampling conditions, it is recommended that samples be collected in regions with high correlations to reduce the uncertainty of CO2 transport analysis due to unknown heterogeneity. © 2024 Society of Chemical Industry and John Wiley & Sons, Ltd.

A comprehensive study on seismic dynamic responses of stochastic structures using sparse grid-based polynomial chaos expansion

The seismic response is a critical aspect of structural engineering, as it directly influences the design and safety evaluation of such systems. Structural seismic response becomes more complicated when unavoidable uncertainties in the structure are considered. Therefore, solving and analyzing the stochastic dynamic response of structures efficiently and accurately is of great significance. Various uncertainty quantification methods such as the stochastic finite element method, Monte Carlo method, and polynomial chaos expansion method have made significant progress in this field. However, these methods do not provide a comprehensive study of the seismic response of stochastic structures and efficiency problems rapidly emerge as the number of random variables increases. This paper delivers a comprehensive study of the seismic dynamic response of space truss and reticulated shell structure considering uncertainties in material properties, geometry, and loading. To this end, the finite element model of the structure was first modeled and the deterministic solutions were obtained. Then, polynomial surrogates are modeled for the natural frequency, mode participation factor, mode superposition response, and transient response of the structure under stochastic uncertainties. The sparse grid technique is used to select sparse quadrature points, which overcomes the curse of dimensionality caused by full grid integration. The results demonstrate that the sparse grid-based polynomial chaos expansion is more efficient. Uncertainty quantification reveals the uncertainty of modal analysis, response spectrum analysis, and transient response analysis of space truss and reticulated shell under seismic excitation. In summary, this study provides valuable insights and an efficient and accurate uncertainty quantification tool for the seismic dynamic analysis of stochastic structures, ultimately contributing to safer and more robust structural designs.

Deterministic and probabilistic analysis of great-depth braced excavations: A 32 m excavation case study in Paris

The Fort d’Issy-Vanves-Clamart (FIVC) braced excavation in France is analyzed to provide insights into the geotechnical serviceability assessment of excavations at great depth within deterministic and probabilistic frameworks. The FIVC excavation is excavated at 32 m below the ground surface in Parisian sedimentary basin and a plane-strain finite element analysis is implemented to examine the wall deflections and ground surface settlements. A stochastic finite element method based on the polynomial chaos Kriging metamodel (MSFEM) is then proposed for the probabilistic analyses. Comparisons with field measurements and former studies are carried out. Several academic cases are then conducted to investigate the great-depth excavation stability regarding the maximum horizontal wall deflection and maximum ground surface settlement. The results indicate that the proposed MSFEM is effective for probabilistic analyses and can provide useful insights for the excavation design and construction. A sensitivity analysis for seven considered random parameters is then implemented. The soil friction angle at the excavation bottom layer is the most significant one for design. The soil-wall interaction effects on the excavation stability are also given.

Investigation on wave-induced responses of heterogeneous seabed around a buried pipeline based on nonstationary random field model

The instability of the seabed poses a threat to the structural integrity and functionality of submarine pipelines. This numerical study investigates wave-induced responses in a heterogeneous seabed around a buried pipeline, considering spatial variability in soil properties through nonstationary random field models. Three scenarios of spatial variability for soil parameters changing with depth are simulated using both stationary and nonstationary random field models via the Karhunen-Loève expansion method. The discrete random fields are validated in terms of correlation structures and statistical characteristics. The stochastic finite element method, incorporating random fields and a poroelastic seabed model through a LiveLink code, is employed to examine probabilistic outcomes of dynamic soil responses and liquefaction zone distributions within a probabilistic framework. Contrastive analyses are conducted for seabed models with different random field scenarios, in addition to deterministic analyses. The results of the stochastic analysis show that the increasing trend of soil strength with depth has significant effects on the seabed responses and transient liquefaction characteristics around a pipeline.

Three-dimensional creep calculation model for reliability analysis of concrete-filled steel tubular (CFST) structure

For long-span concrete-filled steel tube (CFST) structure, it is essential to implement efficient and realistic prediction models for the long-term processes of core concrete creep. In order to systematically study the main influential factors in CFST structure creep effects, a probabilistic analysis can be performed. The objective of this paper is to analyse the long-term response of CFST structure by using time-dependent reliability method. To address the problem of creep calculation, a novel and comprehensive presentation was developed. By converting the integral-type creep law to a differential-type form with internal variables, based on the Kelvin chain model, an elastic structural analysis with generally orthotropic elastic moduli and eigenstrains was established. We have compiled a three-dimensional (3-D) finite-element analysis programme that considers the creep Poisson effect of core concrete to analyse the long-term performance of the CFST structure. This is achieved by developing a commercial finite element code, such as Usermat in ANSYS. As calculation examples, several time-dependent analyses of a CFST stub column and long-span CFST arch bridge were performed. The calculated results of long-term behaviour agree well with experimental and measured data. Considering the randomness of various input parameters, such as physical dimensions and materials of the CFST structure, we have analysed the uncertainty using the Latin-hypercube-sampling (LHS) stochastic finite-element method. The results of creep sensitivity analysis revealed that coefficient of creep model uncertainty, steel and concrete elastic modulus were the three most influential parameters for CFST stub columns’ creep, and load, coefficient of creep model uncertainty, relative humidity, steel and concrete elastic modulus were the five for CFST arch bridges’ long-term deflection. In addition, the use of Monte Carlo (MC) and response surface (RS) methods for structural stochastic reliability analysis demonstrates the ability of these approaches to assess the reliability index of long-span CFST arch bridges. This provides a valuable reference for the design and performance evaluation of CFST structures.

A novel uncertainty quantification method for determining deformations and reliabilities of stochastic laminated composite plates with geometric nonlinearity

Stochastic finite element method (SFEM) for uncertainty quantification is widely applied for analysis of structures with intrinsic randomness. For determining the geometrically nonlinear deformations of laminated composite plates with random fields, the existing intrusive SFEMs have the limitations of low applicability, insufficient accuracy, or low efficiency. To this end, this paper proposes a novel and efficient non-intrusive SFEM incorporating direct probability integral method to achieve probability density functions (PDFs) of stochastic responses and reliabilities of laminated composite plates with geometric nonlinearity. Firstly, the von Kármán strain-displacement relation based on the third-order shear deformation theory is employed to model the geometric nonlinearity of laminated plates. The random field is discretized via Karhunen–Loève expansion. Secondly, the probability density integral equation (PDIE) is derived from the new perspective of probability conservation. The proposed non-intrusive SFEM decouples the equilibrium equation and PDIE to compute the response PDFs and reliabilities of uncertain laminated composite plates in a unified way. Moreover, the criterion which can judge the applicability of geometrically nonlinear theory is suggested for performing uncertainty quantification. Finally, comparisons of the results in terms of Monte Carlo simulation and the literature demonstrate the high accuracy and efficiency of the proposed method. For stochastic laminated plates, the response statistical moments vary nonlinearly with linear increase of load amplitude due to geometric nonlinearity, the deflection variability increases and structural reliability decreases with the increase in variability and correlation length of random field, and the stacking angle significantly affects the stochastic nonlinear deflections and reliabilities.

Stochastic finite element modeling of heterogeneities in massive concrete and reinforced concrete structures

AbstractThe lifespan of a reinforced concrete (RC) structure can be greatly influenced by the spatial variability of its material characteristics which, in particular, explains the observed or measured reduction of the tensile strength at first crack when the volume under tension increases. This paper discusses the ability of accounting for the spatial variability of the tensile strength of concrete in RC structures using a stochastic finite element (SFE) method based on random field simulations. In this work, the generation of random fields on the concrete tensile strength aims at computing the force corresponding to the first crack occurrence, and the reduced tensile strength of the structure. The method can be applied in particular to large‐sized structures, which show a pronounced size effect, for different types of loading. The method consists of, first, estimating the mean of the random field, using the analytical approach of the weakest link and localization method (WL ). Then, the discretized random field is defined on a particular 2D or 3D grid, and it is finally projected on the finite element mesh of the studied structure. The study of the parameters that influence the prediction of the cumulative density functions (CDFs) of the rupture force or the tensile strength is highlighted using experimental series of concrete beams having different volumes and subjected to 4‐point bending loading. Moreover, the SFE method is applied to a RC tie‐beam under tensile loading, characterized by a weak stress gradient, which complicates the prediction of crack positions.

Eigenvibrations of Kirchhoff Rectangular Random Plates on Time-Fractional Viscoelastic Supports via the Stochastic Finite Element Method.

The present work's main objective is to investigate the natural vibrations of the thin (Kirchhoff-Love) plate resting on time-fractional viscoelastic supports in terms of the Stochastic Finite Element Method (SFEM). The behavior of the supports is described by the fractional order derivatives of the Riemann-Liouville type. The subspace iteration method, in conjunction with the continuation method, is used as a tool to solve the non-linear eigenproblem. A deterministic core for solving structural eigenvibrations is the Finite Element Method. The probabilistic analysis includes the Monte-Carlo simulation and the semi-analytical approach, as well as the iterative generalized stochastic perturbation method. Probabilistic structural response in the form of up to the second-order characteristics is investigated numerically in addition to the input uncertainty level. Finally, the probabilistic relative entropy and the safety measure are estimated. The presented investigations can be applied to the dynamics of foundation plates resting on viscoelastic soil.

Finite element method studying the reliability of taper steel frame with semi-rigid connected impact wind load

This article studies the application of the Stochastic Finite Element Method (SFEM) to analyze beveled frames with semi-rigid connections subjected to wind loads when the connection and load parameters are random quantities. The structural behavior analysis algorithm is a combination of the SFEM method and Monte-Carlo simulation. The calculation program was then used to evaluate the reliability of the semi-rigidly connected beveled steel frame subjected to wind loads.

Structural Uncertainty Analysis of High-Temperature Strain Gauge Based on Monte Carlo Stochastic Finite Element Method.

The high-temperature strain gauge is a sensor for strain measurement in high-temperature environments. The measurement results often have a certain divergence, so the uncertainty of the high-temperature strain gauge system is analyzed theoretically. Firstly, in the conducted research, a deterministic finite element analysis of the temperature field of the strain gauge is carried out using MATLAB software. Then, the primary sub-model method is used to model the system; an equivalent thermal load and force are loaded onto the model. The thermal response of the grid wire is calculated by the finite element method (FEM). Thermal-mechanical coupling analysis is carried out by ANSYS, and the MATLAB program is verified. Finally, the stochastic finite element method (SFEM) combined with the Monte Carlo method (MCM) is used to analyze the effects of the physical parameters, geometric parameters, and load uncertainties on the thermal response of the grid wire. The results show that the difference of temperature and strain calculated by ANSYS and MATLAB is 1.34% and 0.64%, respectively. The calculation program is accurate and effective. The primary sub-model method is suitable for the finite element modeling of strain gauge systems, and the number of elements is reduced effectively. The stochastic uncertainty analysis of the thermal response on the grid wire of a high-temperature strain gauge provides a theoretical basis for the dispersion of the measurement results of the strain gauge.

Resonance analysis of a high-speed railway bridge using a stochastic finite element method

Resonance analysis of a high-speed railway bridge using a stochastic finite element method

Static homotopy response analysis of structure with random variables of arbitrary distributions by minimizing stochastic residual error

The modelling of realistic engineering structures with uncertainties often involves various probabilistic distribution types, which bring forward higher requirements for the generality of stochastic analysis methods. This paper proposes a novel stochastic homotopy method to evaluate the static response of the structure with random variables of arbitrary distributions. In this method, a homotopy series expansion is used to approximate the stochastic static response, and the optimal form of the expansion can be determined by minimizing the residual error about the stochastic static equilibrium equation regardless of distribution types of random variables. The numerical results of a logarithmic function and a thin plate show that the new method exhibits excellent accuracy and stability compared to the homotopy stochastic finite element method depending on the sample selection. Compared with the arbitrary polynomial chaos method (aPC), the proposed method is more efficient under equivalent accuracy. On the other hand, as the expansion order increases, this new method shows better convergence than the aPC method and the perturbation stochastic finite element method in the case of non-Gaussian distributed random variables of large fluctuation. In addition, a cable-stayed bridge example illustrates the application of the proposed method on the large-scale structure.