Stochastic Finite Element Research Articles

Cross-mode stabilized stochastic shallow water systems using stochastic finite element methods

The development of surrogate models to study uncertainties in hydrologic systems requires significant effort in the development of sampling strategies and forward model simulations. Furthermore, in applications where prediction time is critical, such as prediction of hurricane storm surge, the predictions of system response and uncertainties can be required within short time frames. Here, we develop an efficient stochastic shallow water model to address these issues. To discretize the physical and probability spaces we use a Stochastic Galerkin method and an Incremental Pressure Correction Scheme to advance the solution in time. To overcome discrete stability issues, we propose cross-mode stabilization methods which employs existing stabilization methods in the probability space by adding stabilization terms to every stochastic mode in a modes-coupled way. We extensively verify the developed method for both idealized shallow water test cases and hindcasting of past hurricanes. We subsequently use the developed and verified method to perform a comprehensive statistical analysis of the established shallow water surrogate models. Finally, we propose a predictor for hurricane storm surge under uncertain wind drag coefficients and demonstrate its effectivity for Hurricanes Ike and Harvey.

Influence of material uncertainties on thermo-elastic vibration characteristics of graphene reinforced functionally graded porous beams

This paper aims to study the influence of material uncertainties on thermo-elastic vibration response of functionally graded porous (FGP) beams reinforced with graphene platelets (GPLs) using the perturbation-based stochastic finite element method. The influence of material uncertainty on the vibration behavior of FGP-GPL reinforced beams subjected to thermal loading conditions have been discussed for low variability (randomness) in material design parameters (porosity content, amount of nanofillers) and material properties (Young’s modulus, density of metal matrix, and nanofillers) respectively. A C0 continuous stochastic finite element formulation based on higher-order shear deformation theory has been formulated. The convergence and validation of the present formulation have been compared with a traditional Monte-Carlo simulation to establish the accuracy of the present methodology. The homogenized effective material properties are estimated using the Halpin-Tsai micromechanics model and Voigt’s rule of mixture and are typically assumed to be varying along the thickness direction. The results highlighting the effect of uncertainty in material properties on the thermo-elastic vibration response of FGP-GPL reinforced beams have been discussed thoroughly.

Local bearing capacity of steel beam webs with random pit corrosion

H-shaped steel beams are widely used in architectural and industrial structures, such as bridges, offshore platforms, and industrial equipment. However, corrosion seems to be inevitable during the service of steel beams, with pitting being one of the most common corrosion types. The local bearing capacity of the steel girder web is one of the most important factors affecting the stability of steel structures. In this work, the stochastic finite element method is used to reveal the effect of pit corrosion on the local bearing capacity of the corroded H steel girder web, and the relationship between the local bearing capacity of the web and the mass loss rate is also discussed. The effects of specific and random corrosion positions on the local bearing capacity of webs, as well as random corrosion thickness and constant corrosion thickness on the web thickness, are analyzed. Moreover, the effect of the reduction coefficient on the probability distribution model is studied, and the variation of the reduction coefficient with the probability distribution model of corrosion thickness is revealed. The conclusions in this work can provide a theoretical analytical basis for the evaluation of the local bearing capacity of H-shaped steel beams with randomly corroded webs.

Axial compression capacity of rectangular CFST columns under random pitting corrosion

To study the effect of random pitting corrosion, this paper presents the change law of the bearing capacity of the rectangular concrete-filled steel tube (CFST) column under the action of axial force with the mass loss ratio of the outer steel tube. Existing experimental data, including the axial ultimate bearing capacity and load displacement curves of the uncorroded and corroded rectangular CFST columns, are collected to verify the rationality of the finite element model. ANSYS software was used to establish a model that can simulate the pitting corrosion, and the results of the finite element analysis were compared with the experiment, which illustrates that the established model can accurately predict the actual load changes. For further research, the corroded thickness tc/t, which include constant tc/t and random tc/t, and other influencing factors such as corrosion pit diameter Dc, steel tube wall thickness t, material strength, shape and section size of specimens are discussed, the results reveal the effect of these parameters on the decline of axial compressive load capacity. Meanwhile, to visually reflect this effect, this study introduces the concept of bearing capacity reduction coefficient Rcs and Rc to express the ratio of bearing capacity of specimens before and after corrosion. The extensive stochastic finite element analysis was carried out, and on this basis, a suitable calculation formula for the reduction factor is proposed to predict the axial compression bearing capacity of the corroded rectangular CFST column by measuring the value of local mass loss ratio of the outer steel.

Efficient structural reliability analysis via a weak-intrusive stochastic finite element method

This paper presents a novel methodology for structural reliability analysis by means of the stochastic finite element method (SFEM). The key issue of structural reliability analysis is to determine the limit state function and corresponding multidimensional integral that are usually related to the structural stochastic displacement and/or its derivative, e.g., the stress and strain. In this paper, a novel weak-intrusive SFEM is first used to calculate structural stochastic displacements of all spatial positions. In this method, the stochastic displacement is decoupled into a combination of a series of deterministic displacements with random variable coefficients. An iterative algorithm is then given to solve the deterministic displacements and the corresponding random variables. Based on the stochastic displacement obtained by the SFEM, the limit state function described by the stochastic displacement (and/or its derivative) and the corresponding multidimensional integral encountered in reliability analysis can be calculated in a straightforward way. Failure probabilities of all spatial positions can be obtained at once since the stochastic displacements of all spatial points have been known by using the proposed SFEM. Furthermore, the proposed method can be applied to high-dimensional stochastic problems without any modification. One of the most challenging problems encountered in high-dimensional reliability analysis, known as the curse of dimensionality, can be circumvented with great success. Three numerical examples, including low- and high-dimensional reliability analysis, are given to demonstrate the good accuracy and the high efficiency of the proposed method.

A stochastic finite element scheme for solving partial differential equations defined on random domains

This paper proposes a novel stochastic finite element scheme to solve partial differential equations defined on random domains. A geometric mapping algorithm first transforms the random domain into a reference domain. By combining the mesh topology (i.e. the node numbering and the element numbering) of the reference domain and random nodal coordinates of the random domain, random meshes of the original problem are obtained by only one mesh of the reference domain. In this way, the original problem is still discretized and solved on the random domain instead of the reference domain. A random isoparametric mapping of random meshes is then proposed to generate the stochastic finite element equation of the original problem. We adopt a weak-intrusive method to solve the obtained stochastic finite element equation. In this method, the unknown stochastic solution is decoupled into a sum of the products of random variables and deterministic vectors. Deterministic vectors are computed by solving deterministic finite element equations, and corresponding random variables are solved by a proposed sampling method. The computational effort of the proposed method does not increase dramatically as the stochastic dimension increases and it can solve high-dimensional stochastic problems with low computational effort, thus the proposed method avoids the curse of dimensionality successfully. Four numerical examples are given to demonstrate the good performance of the proposed method.

Fuzzy finite element analysis based on the transformation between fuzzy and random variables

In this paper, a fuzzy finite element method (FFEM) for determining the responses of structures is proposed by using the transformation between fuzzy and random variables. Firstly, the formulae for establishing normal random variables equivalent to symmetric triangular fuzzy numbers are presented based on the combination of the principle of insufficient reason and that of maximum specificity. As a result, fuzzy finite element analysis is transferred into stochastic finite element analysis. To solve this problem, the response surface method with the aid of standard normal random variables is utilized to approximate the real responses of structures. Then, the errors between the training and test sets are estimated to select a suitable response surface model amongst the regression models. Lastly, the formulae for determining the mean and the standard deviation values of the responses of structures are established. The accuracy and effectiveness of the proposed method are verified via an illustrative example.

Dynamic Response and Reliability Analysis of Stochastic Multi-Story Frame Structures under Random Excitation

In earthquake engineering problems, uncertainty exists not only in the seismic excitations but also in the structure's parameters. This study investigates the influence of structural geometry, elastic modulus, mass density, and section dimension uncertainty on the stochastic earthquake response of a multi-story moment resisting frame subjected to random ground motion. The North-south component of the Ali Gharbi earthquake in 2012, Iraq, is selected as ground excitation. Using the power spectral density function (PSD), the two-dimensional finite element model of the moment resisting frame's base motion is modified to account for random ground motion. The probabilistic study of the moment resisting frame structure using stochastic finite element utilizing Monte Carlo simulation was presented using the finite element program ABAQUS. The dynamic reliability and probability of failure of the stochastic and deterministic structure based on the first passage failure were checked and evaluated. Results revealed that the probability of failure increased due to randomness in stiffness and mass of the structure. Generally, natural frequencies for the lower modes of vibration and relative displacements for the lower stories were more sensitive to the randomness in system parameters.

Strain energy density as a Gaussian process and its utilization in stochastic finite element analysis: Application to planar soft tissues

Data-based approaches are promising alternatives to the traditional analytical constitutive models for solid mechanics. Herein, we propose a Gaussian process (GP) based constitutive modeling framework, focusing on planar, hyperelastic and incompressible soft tissues. Specifically, the strain energy density of soft tissues is modeled as a GP, which can be regressed to experimental stress–strain data obtained from biaxial stretching experiments. Moreover, the GP model can be weakly constrained to be convex. A key advantage of a GP-based model is that, in addition to the mean value, it provides a probability density (i.e. associated uncertainty) for the strain energy density. To simulate the effect of this uncertainty, a non-intrusive stochastic finite element analysis (SFEA) framework is proposed. The proposed framework is verified against an artificial dataset based on the Gasser–Ogden–Holzapfel model and applied to a real experimental dataset of a porcine aortic valve leaflet tissue. The results show that the proposed framework can be trained with limited experimental data and fits the data better than several existing models. The SFEA framework provides a straightforward way of using the experimental data and quantifying the resulting uncertainty in simulation-based predictions.

Stochastic analysis on the lateral buckling of a HPHT pipeline considering the spatial variability of seabed

Complex geological processes and depositional environments could bring great spatial variability of the engineering properties of seabed sediments, which may further cause uncertainty in the global buckling predictions of high-pressure high-temperature (HPHT) pipelines. Unlike previous investigations with the deterministic method, a stochastic finite element (SFE) model is established for predicting the lateral buckling of a pipeline considering the spatial variability of the seabed. A flow chart of the Monto Carlo simulation for the lateral buckling of pipeline is provided. The SFE results for the simplified homogeneous seabed are compared with the existing analytical solutions for model verifications. Monte Carlo simulation results indicate that the mean values of the critical temperature rise for triggering the lateral buckling of a pipeline on the spatially varied seabed are generally larger than the deterministic predictions. It is found that the critical temperature rise may reach its maximum value when the horizontal scale of fluctuation ranges from the value of the discrete-element size to that of the imperfection wavelength. As the horizontal scale of fluctuation increases, the buckling probability increases significantly, which could be up to about 50% while the horizontal scale of fluctuation approaching a quite large value or even infinity.

A Modeling Method for Finite Element Analysis of Corroded Steel Structures with Random Pitting Damage

Aiming at simulating the surface morphology of corroded steel and providing a modeling method with higher accuracy, the accelerated corrosion test was used to obtain six groups of corroded specimens, and then applied to stochastic finite element analysis (FEA) for studying the mechanical behavior of corroded steel. The pitting parameters (the depth, width, and diameter–depth ratio) of all specimens were investigated and statistically analyzed. Considering the irregularity of corroded surface, the random pitting model (RPM) was established based on the secondary development of ABAQUS. Moreover, the rough surface meshing method (RSMM) was subsequently proposed to optimize the element quality of the FEA model. At last, the modeling method was applied to investigate the bearing capacity of corroded steel beams. The results indicate that, firstly, the pitting parameters of all specimens obeyed log-normal distribution, and their logarithmic mean values grew with increase in corrosion time. The corroded surface the RPM can reproduce the evolution behaviors of a corroded surface with higher accuracy. In addition, the FEA model of corroded steel structures can be meshed easily into hexahedron elements by using the RSMM and effectively optimizing the number and quality of elements. By comparing with other test results, the calculation results of the FEA model of steel beams established by using the modeling method proposed in this study demonstrate a good accuracy in mechanical behavior analysis. The modeling method provides further support for the study of mechanical properties of corroded steel structures.

Stochastic isogeometric buckling analysis of composite shell considering multiple uncertainties

An efficient stochastic isogeometric analysis (SIGA) framework considering multiple uncertainties is proposed for the stochastic buckling analysis of composite shells. The framework develops a seamless pipeline enabling uncertainty quantification and uncertainty analysis with the same building blocks of Non-Uniform Rational B-Splines (NURBS). The Gaussian random fields with spatial variability of random parameters are represented by Karhunen-Loève expansion, which is a linear combination of a set of independent random variables and orthogonal functions. The orthogonal functions and corresponding eigenvalues are obtained by solving the Fredholm integral equation of the second kind by the Galerkin isogeometric method. This method exploits the regularity of NURBS basis functions delivering globally smooth eigensolutions and the total mean squared error of the random field is minimized simultaneously. In addition, perturbation technique is developed to solve the stochastic linear buckling equation and predict the second-moment statistics of the buckling load and the probability of failure of structures. The accuracy of the proposed method is assessed by independent Monte Carlo simulation (MCS) based on isogeometric analysis (IGA). Moreover, from the point of view of convergence and computational time, it is evident that the proposed SIGA framework can effectively obtain probabilistic characteristics compared with stochastic finite element method (SFEM).

3D mesoscale analysis of concrete containing defect damages during different freeze-thaw cycles

The damage process of concrete containing air pore and aggregate gradation defects under freeze-thaw (FT) cycles was parametrically studied by an ABAQUS coupled thermo-mechanical mesoscale numerical analyses platform. It was a stochastic finite element model (SFEM) developed by Python script program. The effectiveness of mesoscale model was verified by comparing 96 concrete samples divided to 4 groups with air pore and aggregate defects exposed to 0 to 90 FT cycles. The damage function of concrete for FT cycles was described by Weibull distribution equations. The analyses showed that air pore defects had a significant impact on concrete mechanical characteristics under FT cycles. When FT cycle number (n) reached 90, the strength loss of specimens with no defect was about 37.5%. When air pore content was 5% with non-uniform grading of aggregate, the strength loss of specimen was 61.6%. This showed that FT cycle damage effect on concrete was obvious which was more prominent for higher values of n.

Evaluation of Loess Collapsibility Based on Random Field Theory in Xi’an, China

The engineering properties of collapsible loess have significant uncertainty. Accurate prediction of collapsible deformation is crucial for the safety of engineering construction in loess areas. Taking the typical collapsible loess stratum as the research object in Xi’an, based on the random field theory, combined with the Monte Carlo strategy and modulus reduction method, the stochastic finite element analysis of loess self-weight collapsibility is carried out to study the influence of the spatial variability of compression modulus on the self-weight collapsibility of loess. The results show that the loess tends to be stratified and average along the depth direction with the increase of transverse correlation distance. The random field result of self-weight collapsibility considering the spatial variability of compression modulus is significantly greater than the deterministic result of layered average and the calculated value of loess code. Considering the low compression modulus dominance effect of compression modulus with positive skewed distribution of random field, the equivalent characteristic value of the compression modulus calculated by the layered average modeling for the collapsibility evaluation of typical loess strata in Xi'an area is proposed.

Reliability Analysis of an Uncertain Single Degree of Freedom System Under Random Excitation

In practical engineering problems, uncertainty exists not only in external excitations but also in structural parameters. This study investigates the influence of structural geometry, elastic modulus, mass density, and section dimension uncertainty on the stochastic earthquake response of portal frames subjected to random ground motions. The North-South component of the El Centro earthquake in 1940 in California is selected as the ground excitation. Using the power spectral density function, the two-dimensional finite element model of the portal frame’s base motion is modified to account for random ground motions. A probabilistic study of the portal frame structure using stochastic finite elements utilizing Monte Carlo simulation is presented using the finite element program ABAQUS. The dynamic reliability and probability of failure of stochastic and deterministic structures based on the first-passage failure were examined and evaluated. The results revealed that the probability of failure increases due to the randomness of stiffness and mass of the structure. The influence of uncertain parameters on reliability analysis depends on the extent of variance in structural parameters.

Investigation of a combined Monte Carlo simulation and Galerkin approach for probabilistic characteristics modelling the dynamic behaviour of the fluid-conveying pipe

ABSTRACT A probabilistic approach is used to compute the probabilistic responses of considered fluid-conveying pipes by combining the Galerkin method and Monte Carlo simulation. Analysing the fluid-conveying pipes is often performed based on the deterministic approaches, but the effect of uncertainty needs to be investigated. The natural frequencies can be represented using probabilistic methods by considering the inherent uncertainty in estimating values. The uncertainty was considered as a random variable. The effects of coefficient of variation (COV) associated with the fluid velocity on the probabilistic characteristics are evaluated. The probability distribution functions of natural frequencies are presented. The measured values were validated by comparing with results of the stochastic finite element method. The results imply that the peak points for the diagrams of the standard deviations of the velocity values gained by the Monte Carlo simulation versus the mean velocities of the related values are equal to the critical fluid velocity.

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

AbstractThe large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed‐form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, apth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing thepth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross‐section column, a 7‐story frame, and a Kiewitt single‐layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.

Modeling of Flexible-Link Manipulators Under Uncertain Parameters Based on Stochastic Finite Element Method

Abstract This paper presents a novel approach to obtain the dynamic model of flexible-link manipulators based on the stochastic finite element method. The links and elements of flexible manipulators are affected by uncertainties. The main sources of uncertainties include the variation of mechanical properties. The present research study conveys the following contributions: (i) modeling the uncertain parameters such as the stiffness of the links as an extension of the finite element method based on stochastic fields and the stochastic finite element method, (ii) numerical method to simulate the dynamic response of flexible manipulation with uncertain parameters in the links based on the Monte Carlo simulation (MCS), and (iii) numerical application of the proposed method to the one-link flexible manipulator and two-link flexible manipulator. Numerical simulations illustrate the proposed approach in terms of joint responses and frequency response functions subject to uncertain parameters.

Buckling Analysis of Cylindrical Shells using Stochastic Finite Element Method with Random Geometric Imperfections

AbstractThin‐walled cylindrical shells often exhibit buckling failure and the experimental buckling load is usually lower than calculation results from classical theory and simulation without geometrical imperfection. Besides, test results with carefully conducted similar specimens still have substantial scatter due to imperfection sensitivity. The nonlinear analysis with FEM can obtain a high‐precision result comparing the experiment. However, this is almost impossible in practical engineering. The initial geometric imperfections of cylindrical shells are complex and random properties. Theoretically, these geometric imperfections can be described using a random field or a Fourier series representation. After that a random sample of the buckling bearing capacity of the cylindrical shell can be obtained using the stochastic finite element method within the subsequent analysis. This paper presents the investigation of buckling analysis of cylindrical shells under axial compression considering the randomness of initial geometric imperfections, and the stochastic FEM is employed to calculate the statistical sample of the shell‐buckling load based on a series of random fields and Fourier series representation of geometric imperfections.

Three-scale modeling and probabilistic progressive damage analysis of woven composite laminates

This article investigates the three-scale mechanical response and damage progression of woven composite laminates by probabilistic progressive damage analysis. Micro-scale, meso-scale, and macro-scale stochastic finite element models of woven composite laminates were established to consider the random fiber distribution in yarns and stochastic nature of material properties. Based on Monte-Carlo method, three-scale probabilistic progressive damage model was proposed to predict damage progression and probabilistic distribution of strengths for woven composite laminates. Good agreement between the predictions of elastic properties and strengths and experimental results has been achieved, demonstrating the validity of proposed three-scale probabilistic progressive damage model.