Stochastic Finite Element Formulation Research Articles

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.

Influence of Material Uncertainty on Vibration Characteristics of Higher-Order Cracked Functionally Gradient Plates Using XFEM

In this study, the influence of material uncertainty on the vibration characteristics of the cracked functionally graded materials (FGM) plates is investigated. Extended stochastic finite element formulation is implemented to model the cracked FGM plate with material uncertainty using higher-order shear deformation theory (HSDT). The level set function is employed to track the crack in the FGM domain. The concept of partition of unity technique is implemented to enrich the primary variable with additional functions. The gradation of the material properties along the thickness direction is done using the power-law distribution. The first-order perturbation technique (FOPT) is incorporated in the methodology for stochastic vibration analysis. The convergence and validation study has been performed to verify the efficacy and accuracy of the formulation. Numerical results are obtained to show the effects of various influential parameters like crack length, gradient index, thickness ratio, and boundary condition on the covariance of the square of natural frequencies. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of cracked FGM plates with material randomness.

A static analysis of nonuniform column by stochastic finite element method using weighted integration approach

In general, the fluctuation of the elastic modulus of materials is crucial in structural analysis. This paper develops a stochastic finite element method (SFEM) for analyzing a nonuniform column considering the random process in elastic modulus. This random process of elastic modulus is assumed as a one-dimensional Gaussian random field. The weighted integration method is used to discretize the random field and establish the stochastic finite element formulation to compute the first and second moments of displacement fields. The results of the proposed approach are validated with those of the previous study. The response variability of displacement of column and effect of the parameter of the random field is investigated in detail.

Uncertainty propagation analysis in free vibration of uncertain composite plate using stochastic finite element method

Material uncertainty is more widespread in composite material than the other engineering materials. This uncertainty makes response of these types of structures to be nondeterministic. In order to predict structural reliability, uncertainty in structural responses must be quantified. There is not a reported research in the literature studing free vibration of composite plate with spatially stochastic material properties. In this research, physical and mechanical properties of composite plate including tensile and shear modulus and density of the plate are modeled as stochastic Gaussian fields. Assuming exponential auto covariance kernels for aforementioned stochastic fields, they are discretized to two parts, including deterministic and stochastic parts employing Karhunen-Loeve theorem. Assuming linear form of strains, mechanical strains are defined applying first order shear deformation theory. Kinetic and potential energy of the composite plate is extracted using finite element formulation. Stochastic finite element formulation is derived employing Hamilton’s principle and Euler-Lagrange and equations are verified with the results in the literature for deterministic case. After verification of formulation, material uncertainty effects on uncertainty of natural frequencies are investigated using Monte Carlo simulation. Results show that there is a linear relation between coefficient of variation of uncertain properties and coefficient of variation of stochastic natural frequencies.

Influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates

Present study aims to investigate the influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates. Reddy's higher order shear deformation theory has been employed to model the displacement field. Variational approach has been used to derive the governing differential equations. Effect of material uncertainties in the formulation have been incorporated using first order perturbation technique (FOPT). An efficient stochastic finite element formulation (SFEM) have been used for the calculation of first and second order statics of natural frequency and transverse deflection. Validation of the results have been performed with the help of available literature and separately developed Monte Carlo formulation (MCS) algorithm. A large number of examples have been solved to quantify the effect of uncertainties on the vibration and bending characteristics of functionally graded skew sandwich plates.

Dimension adaptive finite difference decomposition using multiple sparse grids for stochastic computation

The present study aims to investigate uncertainty quantification followed by reliability analysis of structure with homogeneous non-normal random fields. In stochastic finite element formulation, these continuous fields are discretized by different methods (e.g. Karhunen-Loève Expansion) which transformed it into a set of random variables. However, this discretization often leads to large number of random variables, especially for multiple random fields. With this in view, two different meta-model based approaches are presented in this study using high dimensional model representation (HDMR) for efficient stochastic computation. First, an adaptive multiple finite difference HDMR (AMFD-HDMR) is proposed that decomposes the original performance function into summands of smaller dimensions. These subfunctions are modeled by polynomial chaos expansion (PCE) using moving least square technique which utilizes the benefits of orthogonality of the basis functions and provides adaptive interpolation between the support points. These support points are generated in a sparse grid framework based on the hierarchial tensor products of the sub-grids and the appropriate statistical properties. An iterative scheme is developed with the aim to create new support points in the desired locations such as most probable failure point and/or maxima/minima. Later, a dimension adaptive multiple finite difference HDMR (dAMFD-HDMR) is proposed utilizing sensitivity analysis to further improve the efficiency and accuracy. In the second proposal, an intermittent HDMR formulation is suggested based on the individual and mutual contributions of the significant dimensions. Once the meta-model is built, Monte Carlo simulation is performed over it, thus bypassing the time exhaustive computation of the original performance function. Numerical studies are carried out using composite plate to prove the merits of the proposed algorithms compared to other methods available in the literature.

Stochastic finite element on vibration analysis of composite plates having random damping parameters

Damping parameters of composite structures possess significant uncertainty due to the structural complexity of such materials. Considering the parameters as random variables, this work uses the generalized polynomial chaos (gPC) expansion to capture the uncertainty in the damping and in the vibration responses of structures. A non-sampling based stochastic finite element formulation for damped vibration analysis is developed in which the gPC expansion is used to represent damping uncertainty and stochastic responses with unknown deterministic functions. The constructed gPC expansions for the parameters are used as random inputs to the FEM model to realize the responses on a few number of collocation points generated in random space. The realizations then are employed to estimate the unknown deterministic functions of the gPC expansion approximating the responses. The application of the proposed method is practiced on sample fiber-reinforced composite plates having random modal damping parameters. Utilizing a few random collocation points, the method indicates a very good agreement compared to the sampling-based Monte Carlo simulations with large number of realizations.

Stochastic finite element method for random harmonic analysis of composite plates with uncertain modal damping parameters

Damping parameters of fiber-reinforced composite possess significant uncertainty due to the structural complexity of such materials. Considering the parameters as random variables, this paper uses the generalized polynomial chaos (gPC) expansion to capture the uncertainty in the damping and frequency response function of composite plate structures. A spectral stochastic finite element formulation for damped vibration analysis of laminate plates is employed. Experimental modal data for samples of plates is used to identify and realize the range and probability distributions of uncertain damping parameters. The constructed gPC expansions for the uncertain parameters are used as inputs to a deterministic finite element model to realize random frequency responses on a few numbers of collocation points generated in random space. The realizations then are employed to estimate the unknown deterministic functions of the gPC expansion approximating the responses. Employing modal superposition method to solve harmonic analysis problem yields an efficient sparse gPC expansion representing the responses. The results show while the responses are influenced by the damping uncertainties at the mid and high frequency ranges, the impact in low frequency modes can be safely ignored. Utilizing a few random collocation points, the method indicates also a very good agreement compared to the sampling-based Monte Carlo simulations with large number of realizations. As the deterministic finite element model serves as black-box solver, the procedure can be efficiently adopted to complex structural systems with uncertain parameters in terms of computational time.

A random fatigue of mechanize titanium abutment studied with Markoff chain and stochastic finite element formulation

To measure fatigue in dental implants and in its components, it is necessary to use a probabilistic analysis since the randomness in the output depends on a number of parameters (such as fatigue properties of titanium and applied loads, unknown beforehand as they depend on mastication habits). The purpose is to apply a probabilistic approximation in order to predict fatigue life, taking into account the randomness of variables. More accuracy on the results has been obtained by taking into account different load blocks with different amplitudes, as happens with bite forces during the day and allowing us to know how effects have different type of bruxism on the piece analysed.

Stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty

SUMMARYThe problem of representing random fields describing the material and boundary properties of the physical system at discrete points of the spatial domain is studied in the context of linear stochastic finite element method. A randomly parameterized diffusion system with a set of independent identically distributed stochastic variables is considered. The discretized parametric fields are interpolated within each element with multidimensional Lagrange polynomials and integrated into the weak formulation. The proposed discretized random‐field representation has been utilized to express the random fluctuations of the domain boundary with nodal position coordinates and a set of random variables. The description of the boundary perturbation has been incorporated into the weak stochastic finite element formulation using a stochastic isoparametric mapping of the random domain to a deterministic master domain. A method for obtaining the linear system of equations under the proposed mapping using generic finite element weak formulation and the stochastic spectral Galerkin framework is studied in detail. The treatment presents a unified way of handling the parametric uncertainty and random boundary fluctuations for dynamic systems. The convergence behavior of the proposed methodologies has been demonstrated with numerical examples to establish the validity of the numerical scheme. Copyright © 2014 John Wiley & Sons, Ltd.

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

Uncertain Out-of-Plane Behavior of Composite Plates due to Spatially Random Material Properties

This paper deals with the evaluation of response variability of composite plates due to randomness in the Poisson’s ratio in particular for the out-of-plane behavior. To this end, we adopt the weighted integral stochastic finite element (SFE) scheme and derive a SFE formulation employing Taylor’s expansion on the random Poisson’s ratio and displacement vector as well. In case of composite plates, the stiffness of the structure is affected by the stacking scheme of individual layers. This means that even the plates subjected only to in-plane loading can show out-of-plane behavior. In order to investigate this phenomenon, we chose two example plates with symmetric and asymmetric stacking scheme. A detailed explanation on the results is addressed for the plates under consideration.

Plate response variability due to triple random parameters

In this paper, we aim to propose a stochastic finite element formulation having a capability of considering triple random parameters, i.e. elastic modulus, Poisson’s ratio and thickness of plate, simultaneously in the evaluation of response variability of plates. The effects of cross-correlations between the three random parameters on the response variability are also observed. The generalized auto- and cross-correlation functions that are needed in the evaluation of response covariance are derived based on the general expression for the n-th moment of random variables. In order to demonstrate the adequacy of the proposed scheme, we employ a square plate with various boundary conditions as an example, and the results are compared with those given in the literature as well as with the results of Monte Carlo analysis.

Spectral stochastic finite element analysis for laminated composite plates

A spectral stochastic finite element formulation with consideration of multi-layer effect and spatial variability of material properties is developed for probabilistic analysis of laminated composite plates. The material properties of each lamina are modeled as a set of random fields and represented by the Karhunen–Loève expansion. The expansion is incorporated into a spatial discretization in accordance with a standard finite element procedure based on the first-order shear deformation theory. A spectral expansion with use of polynomial chaos is employed to represent the stochastic nodal displacements in terms of standard normal random variables. A preconditioning matrix is then proposed for the solution of spectral stochastic finite element equations with use of a preconditioning conjugate gradient technique. The various statistics of interest of the system responses are finally obtained by means of the coefficients of the spectral expansion. The numerical accuracy and the computational efficiency of the method are demonstrated by comparison with Monte-Carlo simulation.

Acoustic propagation in uncertain waveguides with spatially varying wave speeds

Acoustic propagation in the littoral is a complex process involving acoustic interaction with the ocean surface, the ocean bottom, volume scattering as well as variable propagation characteristics throughout the volume. When modeled stochastically this problem has been approached using a variety techniques including the Monte Carlo method, Neumann expansions and the stochastic finite element method. The last of these has found considerable use because it offers computational efficiencies over the Monte Carlo method and fewer modeling restrictions than the Neumann expansions. This presentation will discuss the development of a one-dimensional stochastic finite element formulation that accounts for spatial variability in wave speed, an area of intense interest not currently addresses in the literature.

Generalized buckling analysis of laminated plates with random material properties using stochastic finite elements

A generalized layer-wise stochastic finite element formulation is developed for the buckling analysis of both homogeneous and laminated plates with random material properties. The pre-buckled stresses are considered in the derivation of geometric stiffness matrix and the effect of variation in these stresses on the mean and coefficient of variation of buckling strength is studied. The mean buckling strength of plates under uniform stress assumption exactly matches with those reported in the literature. However, it is shown that the actual mean buckling strength of plates can be significantly different based on the pre-buckled stress analysis which depends on boundary constraints, principal material directions, aspect and thickness ratios of plates. The statistics of buckling strength is determined using a Taylor series expansion based mean centered first order perturbation technique. The stochastic finite element solutions obtained using layer-wise plate theory is also validated with analytical solutions presented in this paper. Parametric studies are conducted for different aspect ratios, ply orientations and boundary conditions.

On probabilistic viscous incompressible flow of some composite fluids

The perturbation-based stochastic finite element formulation for the viscous incompressible fluid flow with heat transfer is proposed below. Analyzed viscoelastic fluid contains elastic spherical particles with randomly treated radii and total number in the fluid volume; these random variables are defined using their expected values and variances. Starting from these parameters and statistically described fluid viscosity, the probabilistic moments of the effective viscosity for a fluid with these suspensions are derived thanks to the second order perturbation second central probabilistic moment approach. So defined random effective fluid is next studied in the incompressible isobaric Couette flow between parallel plates, where the heat transfer effects are included. Defining boundary velocities and temperatures by their first two moments, the expected values and cross-covariances of the relevant homogenized fluid state functions are calculated. The engineering application of the approach can be modeling of polymers in fluid state (during their processing), reinforced with rubber particles to strengthen the entire composite as well as in general computational simulation of the composite fluids. The stochastic methodology applied to fluids with random solid suspensions can be extended on random flows with stochastic bubbles as well as multiphase coupled flow problems, where partially saturated media are considered [7], too. The main value of the stochastic approach is that it creates the opportunity to determine reliability indices, quite analogously to the research in the domain of solids and structures made of composites.

Transient Dynamics of Stochastically Parametered Beams

The problem of determining the statistics of the transient response of randomly inhomogeneous beams is formulated. This is based on the use of stochastic dynamic stiffness coefficients in conjunction with the fast Fourier transform algorithm. The dynamic stiffness coefficients, in turn, are determined using a stochastic finite-element formulation that employs frequency-dependent shape functions. The approach is illustrated by analyzing the response of a random rod subject to a boxcar type of axial impact and, also, by considering the flexural response of a randomly inhomogeneous beam resting on a randomly varying Winkler's foundation and subjected to the action of a moving force. A discussion on the treatment of system property random fields as being non-Gaussian in nature is presented. Also discussed are the methods for handling nonzero initial conditions within the framework of the frequency domain response analysis employed in the study. Satisfactory comparisons between the analytical results and simulation results are demonstrated.

Green's function for random media

Random coefficients in partial differential equations and boundary conditions pose a computational challenge. The stochastic finite element formulation is involved because the Tatarski convection like terms must be captured via stochastic strain‐displacement matrices. Once the stochastic Green's Function is obtained, standard packages for boundary element analysis, e.g., BEASY, can be employed. Here, for random constitutive properties, a stationary iteration scheme is demonstrated via Fourier transform of distributions. The deterministic Green's function associated with a uniform medium provides the kernel. There is no such analog for stochastic finite elements. In a current bio‐engineering stress analysis program a computer algebra environment, viz. Mathematica, is used to approximate stochastic Green's Functions.

Bimodal bound of system reliability for random composite structures

A procedure of system reliability analysis using the stochastic finite element method for composite structures is presented. Composite structure is considered as a random structure. The material properties, fiber angles, laminate thickness, and curvatures are modeled as random parameters. The curvature means initial curvature as a result of manufacturing error for the case of a flat plate. The random thickness and fiber angles affect the straindisplacement relation and material directions. In the stochastic finite element formulation, these randomnesses are considered to obtain accurate statistical responses. In reliability analysis, composite laminates are assumed to be multicomponent series system, and the correlations between possible failure modes of different components are considered to obtain bimodal bounds of failure probability. The method is verified through comparison of the results with those obtained by the Monte Carlo simulation for two example problems. The developed method is efficient and accurate enough to be applied to engineering problems and can assess the risk of structures quantitatively and logically.