Stochastic Finite Element Method Research Articles

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.

A micromechanical mean‐field homogenization surrogate for the stochastic multiscale analysis of composite materials failure

SummaryThis paper presents the construction of a mean‐field homogenization (MFH) surrogate for nonlinear stochastic multiscale analyses of two‐phase composites that allows the material response to be studied up to its failure. The homogenized stochastic behavior of the studied unidirectional composite material is first characterized through full‐field simulations on stochastic volume elements (SVEs) of the material microstructure, permitting to capture the effect of the microstructural geometric uncertainties on the material response. Then, in order to conduct the stochastic nonlinear multiscale simulations, the microscale problem is substituted by a pressure‐dependent MFH reduced order micromechanical model, that is, a MF‐ROM, whose properties are identified by an inverse process from the full‐field SVE realizations. Homogenized stress‐strain curves can be used for the identification process of the nonlinear range, however, a loss of size objectivity is encountered once the strain softening onset is reached. This work addresses this problematic introducing a calibration of the energy release rate obtained with a nonlocal MFH micromechanical model, allowing to scale the variability found on each SVE failure characteristics to the macroscale. The obtained random effective properties are then used as input of a data‐driven stochastic model to generate the complete random fields used to feed the stochastic MF‐ROM. To show the consistency of the methodology, two MF‐ROM constructed from SVEs of two different sizes are successively considered. The performance of the MF‐ROM is then verified against an experimental transverse‐compression test and against full‐field simulations through nonlocal Stochastic Finite Element Method (SFEM) simulations. The implementation of the energy release rate calibration allows to conduct stochastic studies on the failure characteristics of material samples without the need for costly experimental campaigns, paving the way for more complete and affordable virtual testing.

Stochastic nonlinear eigenvibrations of thin elastic plates resting on time-fractional viscoelastic supports

– The main objective of this work is to investigate natural vibrations of the thin (Kirchhoff-Love) plate resting on time-fractional viscoelastic supports using the Stochastic Finite Element Method (SFEM). The mechanical behavior of the supports is deterministic and is described by the fractional order derivatives of the Riemann-Liouville type. A subspace iteration method in conjunction with the continuation method is the basis of the numerical solution carried out using the displacement formulation of the Finite Element Method. The probabilistic analysis includes independent Monte-Carlo simulation, the semi-analytical approach, and the iterative generalized stochastic perturbation method of the first two probabilistic moments. Moreover, a corresponding relative entropy and the entropy-based reliability index are contrasted with the First Order Reliability Method (FORM). It has been found that the relative entropy for analyzed eigen-vibrations is some interesting alternative to the FORM methodology in reliability assessment.

Investigation of the Eigenvector of Stochastic Finite Element Methods of Functionally Graded Beams with Random Elastic Modulus

This paper presents a stochastic finite element method to calculate the variation of eigenvalues and eigenvectors of functionally graded beams. The modulus of functionally graded material is assumed to have spatial uncertainty as a one-dimensional random field. The formulation of the stochastic finite element method for the functionally graded beam due to the randomness of the elastic modulus of the beam is given using the first-order perturbation approach. This approach was validated with Monte Carlo simulation in previous studies using spectral representation to generate the random field. The statistics of the beam responses were investigated using the first-order perturbation method for different fluctuations of the elastic modulus. A comparison of the results of the stochastic finite element method with the first-order perturbation approach and the Monte Carlo simulation showed a minimal difference.

Random vibration analysis of FGM plates subjected to moving load using a refined stochastic finite element method

The article introduces a finite element procedure using the bilinear quadrilateral element or four-node rectangular element (namely Q4 element) based on a refined first-order shear deformation theory (r-FSDT) and Monte Carlo simulation (MCS), so-called refined stochastic finite element method to investigate the random vibration of functionally graded material (FGM) plates subjected to the moving load. The advantage of the proposed method is to use r-FSDT to improve the accuracy of classical FSDT, satisfy the stress-free condition at the plate boundaries, and combine with MCS to analyze the vibration of the FGM plate when the parameter inputs are random quantities following a normal distribution. The obtained results show that the distribution characteristics of the vibration response of the FGM plate depend on the standard deviation of the input parameters and the velocity of the moving load. Furthermore, the numerical results in this study are expected to contribute to improving the understanding of FGM plates subjected to moving loads with uncertain input parameters.

Stochastic Finite Element Analysis of Plate Structures Considering Spatial Parameter Random Fields

For plate structures, their random parameters can be regarded as a two-dimensional random field in the plane. To solve the plate theory considering a two-dimensional random field, an efficient strategy for the stochastic finite element method was adopted. Firstly, the stochastic finite element method was used to establish the plate structural model, in which the random field characteristics of the parameter were considered, and the mathematical expression of its random field was obtained through the Karhunen–Loève expansion; secondly, the point estimate method was applied to calculate the statistics of random structures. The computational efficiency can be significantly improved through the reference point selection strategy. The accuracy and efficiency of the calculation strategy were verified, and the influences of correlation length and coefficient of variation of the parameter on the random response of plate structures under different plate types (including Kirchhoff plate and Mindlin plate) and boundary conditions (including simply supported and clamped supported) were discussed. The proposed method can provide some help in solving static problems of plate structures.

The a posteriori finite element method (APFEM), a framework for efficient parametric study and Bayesian inferences

Stochastic methods have recently been the subject of increased attention in Computational Mechanics for their ability to account for the stochasticity of both material parameters and geometrical features in their predictions. Among them, the Galerkin Stochastic Finite Element Method (GSFEM) was shown to be particularly efficient and able to provide accurate output statistics, although at the cost of intrusive coding and additional theoretical algebraic efforts. In this method, distributions of the stochastic parameters are used as inputs for the solver, which in turn outputs nodal displacement distributions in one simulation. Here, we propose an extension of the GSFEM—termed the A posteriori Finite Element Method or APFEM—where uniform distributions are taken by default to allow for parametric studies of the inputs of interest as a postprocessing step after the simulation. Doing so, APFEM only requires the knowledge of the vertices of the parameter space. In particular, one key advantage of APFEM is its use in the context of Bayesian inferences, where the random evaluations required by the Bayesian setting (usually done through Monte Carlo) can be done exactly without the need for further simulations. Finally, we demonstrate the potential of APFEM by solving forward models with parametric boundary conditions in the context of (i) metamaterial design and (ii) pitchfork bifurcation of the buckling of a slender structure; and demonstrate the flexibility of its use for Bayesian inference by (iii) inferring friction coefficient of a half plane in a contact mechanics problem and (iv) inferring the stiffness of a brain region in the context of cancer surgical planning.

Semi-reduced order stochastic finite element methods for solving contact problems with uncertainties

This paper develops two-step methods for solving contact problems with uncertainties. In the first step, we propose stochastic Lagrangian multiplier/penalty methods to compute a set of reduced basis. In the stochastic Lagrangian multiplier method, the stochastic solution is represented as a sum of products of a set of random variables and deterministic vectors. In the stochastic penalty method, the problem is divided into the solutions of non-contact and possible contact nodes, which are represented as sums of the products of two different sets of random variables and deterministic vectors, respectively. The original problems are then transformed into deterministic finite element equations and one-dimensional (corresponding to stochastic Lagrangian multiplier method)/two-dimensional (corresponding to stochastic penalty method) stochastic algebraic equations. The deterministic finite element equations are solved by existing numerical techniques, and the one-/two-dimensional stochastic algebraic equations are solved by a sampling method. Since the computational cost for solving stochastic algebraic equations does not increase dramatically as the stochastic dimension increases, the proposed methods avoid the curse of dimensionality in high-dimensional problems. Based on the reduced basis, we propose semi-reduced order Lagrangian multiplier/penalty equations with two components in the second step. One component is a reduced order equation obtained by smooth solutions of the reduced basis and the other is the full order equation for the nonsmooth solutions. A significant amount of computational cost is saved since the sizes of the semi-reduced order equations are usually small. Numerical examples of up to 100 dimensions demonstrate the good performance of the proposed methods.