Stochastic Finite Element Analysis Research Articles

On the performance evaluation of stochastic finite elements in linear and nonlinear problems

Due to the inherent uncertainties in various systems, deterministic approaches may not be able to satisfactorily characterize their response. In such cases, stochastic approaches that can systematically consider uncertainties have to be employed. In the past, many stochastic finite element analysis (SFEA) methods have been developed for uncertainty quantification (UQ), among which the perturbation methods, intrusive and non-intrusive polynomial chaos expansion (IPCE/NIPCE) methods and stochastic collocation (SC) methods have received considerable attention. However, in mechanics, most of the applications of these methods are confined to relatively simple problems, and the applicability and performance of these methods to complex nonlinear mechanics problems are not clear. To this end, this study carried out an investigation on the performance of different SFEA methods in linear and nonlinear problems. Numerical studies show that the NIPCE and SC methods are superior in terms of accuracy among other methods in the linear elastic case. The NIPCE method is also used for UQ in stochastic models with plasticity and nonlocal elastoplastic damage. The results demonstrate that the stochastic averages can be significantly different from the deterministic results, which indicates the necessity of considering UQ for improving response predictions.

Stochastic stress-based topology optimization of structural frames based upon the second deviatoric stress invariant

This work pertains to the stochastic stress-based topology optimization of frame structures considering uncertainty. Specifically, this paper presents an investigation of the second deviatoric stress invariant, J2, as the measure of stress in the domain where the objective is to minimize the maximum of the expected values of the stress invariant. Analytical expressions for the expected value of the J2 invariant are developed employing the perturbation approach. The premise being that the J2 invariant eliminates the square root operator that would otherwise be present if using the von Mises stress, and by eliminating this operator, the nonlinearity in the functional mapping between the random input (magnitude and direction of external forces) and uncertain output (expected values of stress) is reduced. Hence, an improved accuracy can be achieved with an analytical approximation of a given order. The analytical expressions of the expected value of the J2 invariant to the second order are obtained based on a Taylor series expansion along with the associated sensitivities for gradient-based optimization. The analytical expressions are implemented in an optimization scheme and applied for the design of three ground structures considering different variability in the input random variables. For each example, the relative error between the maximum of the expected values of the J2 invariant obtained using the analytical expressions and corresponding values evaluated by performing stochastic finite element analysis, whereby the input distribution is sampled using Monte Carlo methods, finite element simulation performed for each sample realization, then computing the expected value of the stress measure using straightforward statistical expressions, and finally estimation of the maximum via the p-norm, were observed to be less than 2%, even for a coefficient of variation of 0.4. The optimized designs were again analyzed using the von Mises stress and the relative error again computed with respect to those obtained from Monte Carlo sampling and finite element simulation. In general, the relative error using the analytical expressions of the von Mises stress was greater than, except in a couple instances, those obtained using the J2 invariant.

Development of hybrid dimension adaptive sparse HDMR for stochastic finite element analysis of composite plate

A novel adaptive high dimensional model representation (HDMR) is developed for stochastic finite element analysis of composite plate. Uncertainty propagating through the physical system is quantified using this technique where only the significant dimensions are retained. Influence of these dimensions is modeled by different orders which are preferred to be low for computational tractability. These dimensions are identified by a sensitivity analysis using support points in the sparse grid which are generated using multiple levels of hierarchy, joint or marginal probability distributions and dimension adaptiveness. This multi level hierarchical generation of support points fill the domain of the input variables with major emphasis on the regions of interest which may be the entire space, limit state, maxima/minima or tail ends of the probability distribution function for rare events depending upon the nature of the problem. The hybrid characteristics of the proposed dimension decomposition is introduced in this paper where the component functions are expressed using orthogonal bases and the error term which is further modeled using Gaussian random process (i.e. Kriging). The performance of the proposed Hybrid dimension adaptive HDMR (hdA-HDMR) is illustrated using stochastic finite element analysis of laminated composite plate whose material properties are considered as homogeneous non-normal random fields.

Stochastic Finite Element Analysis Framework for Modelling Electrical Properties of Particle-Modified Polymer Composites.

Properties such as low specific gravity and cost make polymers attractive for many engineering applications, yet their mechanical, thermal, and electrical properties are typically inferior compared to other engineering materials. Material designers have been seeking to improve polymer properties, which may be achieved by adding suitable particulate fillers. However, the design process is challenging due to countless permutations of available filler materials, different morphologies, filler loadings and fabrication routes. Designing materials solely through experimentation is ineffective given the considerable time and cost associated with such campaigns. Analytical models, on the other hand, typically lack detail, accuracy and versatility. Increasingly powerful numerical techniques are a promising route to alleviate these shortcomings. A stochastic finite element analysis method for predicting the properties of filler-modified polymers is herein presented with a focus on electrical properties, i.e., conductivity, percolation, and piezoresistivity behavior of composites with randomly distributed and dispersed filler particles. The effect of temperature was also explored. While the modeling framework enables prediction of the properties for a variety of filler morphologies, the present study considers spherical particles for the case of nano-silver modified epoxy polymer. Predicted properties were contrasted with data available in the technical literature to demonstrate the viability of the developed modeling approach.

Developing “Design by Analysis” Methodology for Windows for Pressure Vessels for Human Occupancy

Abstract The ASME pressure vessels for human occupancy (PVHO) codes and standards are engineering standards developed to provide a reliable design method for pressure vessel windows. This empirical method is based primarily on years of government-sponsored testing and development and does not directly use engineering theory. This empirical algorithm makes it challenging to revise without additional large-scale physical testing. The industries using the PVHO code need a way to incorporate advances in material science, manufacturing technology, and overall engineering advances without spending years in code case review. Verification and validation techniques, coupled with stochastic finite element analysis (FEA) to address operational variables, can be the basis for a “design by analysis” method to complement the existing testing requirements to produce a full engineering package consistent with other pressure vessel and pressure vessel component design. A design method sufficiently reliable for PVHO could be used in other applications.

New Discretization Methodology for Three-Dimensional Mechanical Parameters Considering Spatial Variations

In this paper, the uncertain mechanical parameters are modeled as spatial random fields (RFs). A new tetrahedral discretization methodology of three-dimensional (3D) RF is proposed based on the local average (LA) theory. An analytical formula and a numerical formula of the covariance for any two tetrahedral RF elements are developed. Its main advantage, compared to similar studies employing hexahedral discretization method, is that tetrahedral LA method can perfectly combine with tetrahedral finite element (FE) method. Also, the corresponding relation between RF mesh and FE mesh is clearer and the computer codes are simpler. The accuracy of proposed LA method in this study was verified by comparing with other discretization techniques of RF. Finally, an illustrative example is presented to demonstrate the analysis process. The validity and the advantage of the proposed LA method are proven. Taking the spatial variations of uncertain mechanical parameters into account, the stochastic effects of scale of fluctuation on the RF are estimated. The proposed method can be widely applied to 3D stochastic FE analysis, and the sample results will improve our understanding of the stochastic influence of scale of fluctuation on the RF.

Stochastic finite element analysis of rockfill dam considering spatial variability of dam material porosity

PurposeThe purpose of this study is to develop a stochastic finite element method (FEM) to solve the calculation precision deficiency caused by spatial variability of dam compaction quality.Design/methodology/approachThe Choleski decomposition method was applied to generate constraint random field of porosity. Large-scale laboratory triaxial tests were conducted to determine the quantitative relationship between the dam compaction quality and Duncan–Chang constitutive model parameters. Based on this developed relationship, the constraint random fields of the mechanical parameters were generated. The stochastic FEM could be conducted.FindingsWhen the fully random field was simulated without the restriction effect of experimental data on test pits, the spatial variabilities of both displacement and stress results were all overestimated; however, when the stochastic FEM was performed disregarding the correlation between mechanical parameters, the variabilities of vertical displacement and stress results were underestimated and variation pattern for horizontal displacement also changed. In addition, the method could produce results that are closer to the actual situation.Practical implicationsAlthough only concrete-faced rockfill dam was tested in the numerical examples, the proposed method is applicable for arbitrary types of rockfill dams.Originality/valueThe value of this study is that the proposed method allowed for the spatial variability of constitutive model parameters and that the applicability was confirmed by the actual project.

A juxtaposition of data driven and stochastic finite element analyses for problems with noisy material data

AbstractIn data driven modeling the constitutive model is replaced by a data set and therefore yields a model‐free computation. In this context the data set also can be the input for noisy material data and describe material uncertainty. This contribution presents a juxtaposition of different finite element trechniques to describe material uncertainties, namely the stochastic finite element method and a data driven finite element method.

Probabilistic dynamic analysis of composite plates due to low velocity impact

The low velocity impact (LVI) analysis of fiber-reinforced plastic (FRP) plates is a significant study to evaluate the reliability of lightweight structures. This study has wide applications in offshore and naval industries. Safety and reliability assessment as per the international standards is one of the basic objectives of the study. LVI on FRP plates are studied taking the material parameters and loading as random variables. FRP plates are subjected to failure under impact by in-plane loading. To evaluate the safe load carrying capacity and the reliability under impact, dynamic analysis of composite plate subjected to LVI is carried out. Reliability analysis is performed to calculate the stochastic behavior of FRP plates. During impact, the in-plane damage modes such as matrix cracking, fiber failure, and shear cracking are modeled using a failure criterion. The out of plane delamination is modeled using cohesive surfaces. The variability related with the system properties due to the inherent scatter in the geometric and material properties and input loads are modeled in a stochastic fashion. The stochastic finite element analysis (SFEA) is performed to determine the stochastic response of system using Gaussian process response surface method (GPRSM). The safety level qualification is achieved in terms of reliability level targeted.

Stochastic Finite Element Analysis Framework for Modelling Mechanical Properties of Particulate Modified Polymer Composites.

Polymers have become indispensable in many engineering applications because of their attractive properties, including low volumetric mass density and excellent resistance to corrosion. However, polymers typically lack in mechanical, thermal, and electrical properties that may be required for certain engineering applications. Therefore, researchers have been seeking to improve properties by modifying polymers with particulate fillers. In the research presented herein, a numerical modeling framework was employed that is capable of predicting the properties of binary or higher order composites with randomly distributed fillers in a polymer matrix. Specifically, mechanical properties, i.e., elastic modulus, Poisson’s ratio, and thermal expansion coefficient, were herein explored for the case of size-distributed spherical filler particles. The modeling framework, employing stochastic finite element analysis, reduces efforts for assessing material properties compared to experimental work, while increasing the level of accuracy compared to other available approaches, such as analytical methods. Results from the modeling framework are presented and contrasted with findings from experimental works available in the technical literature. Numerical predictions agree well with the non-linear trends observed in the experiments, i.e., elastic modulus predictions are within the experimental data scatter, while numerical data deviate from experimental Poisson’s ratio data for filler volume fractions greater than 0.15. The latter may be the result of morphology changes in specimens at higher filler volume fractions that do not comply with modelling assumptions.

Stochastic finite element analysis framework for modelling thermal conductivity of particulate modified polymer composites

Abstract Polymers possess desirable properties such as low specific gravity and good corrosion resistance, which make them attractive for a variety of applications. However, polymers typically lack in mechanical, thermal and electrical properties. Therefore, considerable research efforts are spent to improve polymer properties using particulate fillers. A multitude of filler materials and shapes are available, such as fiber/rod, plate-like or spherical geometries. Composite morphologies may also differ based on particle distribution, dispersion and alignment, all of which may be affected by filler loading and manufacturing routes. Given the significant efforts associated with experimental analyses, modeling approaches are expedient for providing a fundamental understanding for particulate modified polymers. In the present study a numerical modeling framework was developed for predicting the effective properties, specifically, thermal conductivity, of particulate modified polymers considering filler-matrix and particle-to-particle interfacial effects. Employing a stochastic finite element analysis, composites with randomly distributed spherical particles were modeled and validated against experimental data. The developed modeling framework, which can accommodate a broad range of filler shapes and composite morphologies, provides an effective means for elucidating experimental studies and guiding the design of particulate modified polymer composites.

A novel method of reliability-centered process optimization for additive manufacturing

Process optimization problem of additive manufacturing nowadays is a research hotspot in the field of manufacturing industry. However, parameter uncertainty has not been considered in the past. In this paper, the process optimization methods of additive manufacturing are reviewed, and a novel reliability-centered optimization combining stochastic finite element analysis (SFEA) with particle swarm optimization (PSO) method is proposed and have been explained in details. Finally, the Direct Metal Deposition (DMD) process is used as an example in this paper, and parameters including the layer thickness, heat generation of the melt, scanning speed, as well as the hot bed temperature flux are taking into account. Deformation after the part cools down is the single optimization objective, and the reliability performance is treated as an uncertain constraint in the optimization problem. As a result, the best process parameters of DMD are obtained, and the case study verified the superiority of the proposed method.

Towards an understanding of variations in the buckling of tailored variable angle tow composite plates

In this paper, variable angle tow (VAT) composite plates tailored to enhance buckling performance are studied with the use of stochastic finite element method to quantify uncertainties in buckling measures arising from variations in material properties and fibre tow path. Detailed formulations for predicting buckling statistics in terms of mean value and standard deviation are derived to enable a perturbation-based stochastic finite element analysis. The derivations are built on a linear variation formula for fibre tow path and plate element based on the first order shear deformation theory. They are integrated with Taylor series expansion to propagate uncertainties from inputs to buckling performance measures, including buckling eigenvalues, critical buckling coefficients, etc. A twelve-layer VAT composite plate, with optimally designed fibre tow paths under various boundary conditions, has been investigated to illustrate the uncertainty quantification procedure. The performance of the perturbation-based stochastic finite element method has been validated using Monte Carlo simulation. Influences of variations in material properties and fibre tow path are thoroughly examined to understand the variability of buckling performance of VAT composites.

An explicit method for simulating non-Gaussian and non-stationary stochastic processes by Karhunen-Loève and polynomial chaos expansion

A new method is developed for explicitly representing and synthesizing non-Gaussian and non-stationary stochastic processes that have been specified by their covariance function and marginal cumulative distribution function. The target process is firstly represented in the Karhunen-Loève (K-L) series form, the random coefficients in the K-L series is subsequently decomposed using one-dimensional polynomial chaos (PC) expansion. In this way, the target process is represented in an explicit form, which is particularly well suited for stochastic finite element analysis of structures as well as for general purpose simulation of realizations of these processes. The key feature of the proposed method is that the covariance of the resulting process automatically matches the target covariance, and one only needs to iterate the marginal distribution to match the target one. Three illustrative examples are used to demonstrate the proposed method.

Reliability assessment of glass epoxy composite plates due to low velocity impact

Probability of failure estimation of an offshore composite structure under low velocity impact is an essential requirement to incorporate the uncertainties associated with material properties and load due to an impact. The likelihood of this hazard causing a chain of failure events plays an imperative role in risk assessment. The material properties of composites mostly exhibit a scatter due to their in-homogeneity and anisotropic characteristics, brittleness of the matrix and fibre and manufacturing defects. In reality, the possibility of incidence of such a situation is due to large randomness arising in the structure. Stochastic finite element analysis of fibre reinforced plastic plates under low velocity impact is performed to account for the uncertainties of material properties and initial impact velocity. Impact induced failure of fibre reinforced plastic structures are a stochastic process due to scatter arising in material and impact behaviour, since individual faults in the ply are not easy to track. The continuum damage model is implemented in the finite element code by a user-defined material subroutine to overcome these problems. The Gaussian process response surface method is presently adopted to determine the probability of failure. Comparative study is also carried out for different combination of impactor velocities and mass. The stochastic design optimization method is performed to attain superior strength and lighter weight of fibre reinforced plastic structures.

Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters

• A general approach for the stochastic finite element analysis of soft tissue deformation is presented. • Sensitivity derivative Monte Carlo method to propagate uncertainty through hyperelastic models with stochastic parameters. • The proposed method is up to two orders of magnitude faster than the standard Monte Carlo approach. • The method is able to provide the user with statistical results on quantities of practical interest. • We applied the approach to a simple academic example and to the stochastic deformation of a brain. We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than the standard Monte Carlo method. The material model of interest can be altered by adjusting few lines of (FEniCS) code. The method is able to (1) provide the user with statistical confidence intervals on quantities of practical interest, such as the displacement of a tumour or target site in an organ; (2) quantify the sensitivity of the response of the organ to the associated parameters of the material model. We exercise the approach on the determination of a confidence interval on the motion of a target in the brain. We also show that for the boundary conditions under consideration five parameters of the Ogden–Holzapfel-like model have negligible influence on the displacement of the target zone compared to the three most influential parameters. The benchmark problems and all associated data are made available as supplementary material.

Computational-experimental approaches for fatigue reliability assessment of turbine bladed disks

In the present study, a computational-experimental framework is developed for fatigue reliability assessment of turbine bladed disks. Within the framework, the overspeed testing is innovatively combined with stochastic finite element (FE) analysis for quantifying uncertainties in the experimental data, material properties and loads. Meanwhile, two schemes are elaborated based on probabilistic S-N curves and stochastic FE simulation coupling with sampling technique. The stochastic FE simulation incorporates the Chaboche constitutive model with Fatemi–Socie criterion for fatigue behavior modeling and life prediction. Moreover, experimental deformation and numerical FE analysis are conducted with regard to the full-scale bladed disk test with increased step-stress overloading. Reliability sensitivity analysis is performed to provide an importance ranking of random variables for fatigue design of the bladed disk. Results indicate that stochastic FE analysis-based scheme provides more conservative predictions than the probabilistic S-N curves-based one.

Stochastic finite element response analysis using random eigenfunction expansion

A mathematical form for the response of the stochastic finite element analysis of elliptical partial differential equations has been established through summing products of random scalars and random vectors. The method is based upon the eigendecomposition of a system’s stiffness matrix. The computational reduction is achieved by only summing the dominant terms and by approximating the random eigenvalues and the random eigenvectors. An error analysis has been conducted to investigate the effect of the truncation and the approximations. Consequently, a novel error minimisation technique has been applied through the Galerkin error minimisation approach. This has been implemented by utilising the orthogonal nature of the random eigenvectors. The proposed method is used to solve three numerical examples: the bending of a stochastic beam, the flow through a porous media with stochastic permeability and the bending of a stochastic plate. The results obtained through the proposed random eigenfunction expansion approach are compared with those obtained by using direct Monte Carlo Simulations and by using polynomial chaos.

Stochastic finite element analysis of composite structures based on mesoscale random fields of material properties

The linking of microstructure uncertainty with the random variation of material properties at higher scales is of paramount importance in the framework of the stochastic finite element method (SFEM). An efficient computational scheme has been recently proposed by the authors for the determination of mesoscale random fields for the apparent properties and of the RVE size of particle-reinforced composites based on computer-simulated images of their microstructure. The proposed approach exploits the excellent synergy of the extended finite element method (XFEM) and Monte Carlo simulation in order to analyze the microstructure models and obtain statistical information (probability distribution, correlation structure) for the apparent properties of the composite in each mesoscale size. In this paper, sample functions of the statistically homogeneous mesoscale random fields are generated using the spectral representation method in conjunction with translation field theory. Moreover, micro-mechanically consistent (lower and upper) bounds on the macroscopic response of composite structures are computed in the framework of SFEM. Useful conclusions are derived regarding the effect of particle/matrix stiffness ratio on the probabilistic characteristics of the mesoscale random fields as well as the influence of mesoscale size on the response variability.

Micromechanical investigation and numerical simulation of fatigue crack formation in welded joints

The present study is directed to the nucleation of micro cracks in polycrystalline metals due to fatigue. For this purpose, a micromechanics based numerical procedure is established. This procedure is based on the assumption that in the initial phase, the fatigue crack propagation follows the slip planes of the individual grains. The crack propagation is modelled in terms of a damage mechanics concept, assuming that fatigue damage is driven by the dissipated microplastic work. Using a generalized homogenization approach, the results of the micromechanical simulation are transferred to the macroscopic level. Using a stochastic finite element analysis, the corresponding uncertainty and scatter are assessed. The numerical scheme is validated against micromechanical experimental investigations of the crack formation using micro scale specimens with gauge sections in the range of 200µm×450µm.