Perturbation-based Stochastic Finite Element Method Research Articles

Uncertainty in the Flexural Behavior of Soft-Core Sandwich Beams

AbstractThe impact of uncertainty on the flexural response of sandwich beams is studied in this paper. The study focuses on the unique features of typical soft-core sandwich beams, including local effects and stresses concentrations. Those are critical for the integrity and performance of the structural system and, to the best knowledge of the authors, were never considered from a stochastic viewpoint. Several structural features of the beam are considered, separately, as uncertain, and the stochastic characteristics of the distributions of the displacements and stresses are investigated. To capture the unique local effects, the extended high-order sandwich panel theory is adopted. Numerical solutions of the stochastic ordinary differential equations use the finite element (FE) method. The parametric uncertainty and its spatial distribution are modeled by random fields. The stochastic analysis uses both the Karhunen–Loeve polynomial-chaos and the perturbation-based stochastic finite element methods. The two approaches are compared and comprehensively validated against Monte Carlo simulations. The numerical results quantify the impact of uncertainty on the response, but show significant disagreements between the two stochastic approaches, and the perturbation-based stochastic finite element method is chosen as the most suitable one. The effects of the coefficient of variation of the uncertain input parameter and its correlation length are also studied. Highly amplified levels of uncertainty are revealed by the stochastic analysis near points of local stress concentration. These localized yet significant uncertainties, reported here for the first time, shed new light on the design, analysis, and safety of such sandwich beams.

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.

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.

Influence of stochastic variations in manufacturing defects on the mechanical performance of textile composites

This paper presents a methodology to evaluate the effects of microscopic manufacturing defects, namely fibre misalignment, waviness and volume fraction, on the mechanical performance. Influences of these defects on the effective elastic properties of composites are quantified by a dual homogenization method. For estimating stochastic characteristics of the properties induced by the variations in these defects, a probabilistic extension of the dual homogenization method is developed and numerically implemented through a perturbation-based stochastic finite element method. It is further incorporated in a multiscale finite element based reliability method to measure the influences of these manufacturing defects on structural performance in terms of reliability. The effectiveness of the proposed method in capturing defects is illustrated initially by investigating the effective elastic properties of a unidirectional fibre composite based yarn and then a plain woven textile composite. The capability of the proposed method in quantifying the variations in these defects is further demonstrated through statistical analysis of the effective elastic properties and a woven textile composite and structural reliability analysis of a textile composite laminate. This paper represents a significant advancement in the probabilistic prediction of the behaviour of woven and non-woven composites.

Stochastic seismic analysis of Kömürhan Highway Bridge with varying material properties

Stochastic seismic finite element analyses of the Kömürhan Bridge, the material properties of which are described by random fields, are presented in this paper. The stochastic perturbation technique and Monte Carlo simulation (MCS) method are used in the analyses. A summary of MCS and perturbation-based stochastic finite element dynamic analysis formulation of the structural system is given. The Kömürhan Bridge, located on the 51st km of Elazığ-Malatya highway in the east of Turkey, was chosen as a numerical example. The Erzincan earthquake in 1992 was considered a ground motion, since it took place in the vicinity of the bridge. The material properties were considered to be random variables. During the stochastic analysis, displacements and internal forces of the bridge under consideration were obtained using the perturbation-based stochastic finite element method (SFEM), as well as the MCS method. The selected random variables were elastic modulus and mass density. The efficiency and accuracy of the proposed SFEM algorithm were validated through comparison with results of the MCS method.

2D versus 3D probabilistic homogenization of the metallic fiber-reinforced composites by the perturbation-based stochastic Finite Element Method

The main purpose of this work is computational simulation of the expectations, standard deviations, skewness and kurtosis of the homogenized tensor for some composites with metallic components. The Representative Volume Element (RVE) of this composite contains a single cylindrical fiber and their components are treated as statistically homogeneous and isotropic media uniquely defined by the Gaussian elastic modulus. Probabilistic approach is based on the generalized stochastic perturbation technique allowing for large random dispersions of the input random variables and is implemented using the polynomial response functions recovered using the Least Squares Method. Homogenization technique employed is dual and consists of (1) stress version of the effective modules method and (2) its displacements counterpart based on the deformation energies of the real and homogenized composites. The cell problem is solved for the first case by the plane strain homogenization-oriented code MCCEFF and, in the 3D case, using the system ABAQUS® (8-node linear brick finite elements C3D8), where the uniform deformations are imposed on specific outer surfaces of the composite cell; probabilistic part is carried out in the symbolic computations package MAPLE®. We compare probabilistic coefficients of the effective elasticity tensor computed in this way with the corresponding coefficients for their upper and lower bounds and this is done for the composite with small and larger contrast between Young moduli of the fiber and the matrix. The main conclusion coming from the performed numerical analysis is a very good agreement of the probabilistic moments resulting from 2 and 3D computer models; this conclusion is totally independent from the contrast between elastic moduli of both composite components.

Optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method

The main objective here is an engineering optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method. This procedure is based on triple reliability analysis consisting of the limit states for the admissible eigenfrequencies, horizontal displacements and the reduced stresses; it obeys an application of the stainless steel as well as the aluminum as the structural materials. Such a reliability-based optimization follows strictly engineering practice, demands of the Eurocodes, cost minimization of the structures and future installations of the new transmission equipment. The reliability analysis is based on the FORM approach, while computational realization of the generalized SFEM guarantees reliable determination of the first four probabilistic moments of the structural response for any random dispersion in the design parameters. Numerical implementation has hybrid character and is based on interoperability of the FEM design system ROBOT and the computer algebra system MAPLE, where all probabilistic functions are programmed.

Reliability Modeling in Some Elastic Stability Problems Via the Generalized Stochastic Finite Element Method / Modelowanie Niezawodnosci W Pewnych Zagadnieniach Statecznosci Sprezystej Z Wykorzystaniem Uogólnionej Stochastycznej Metody Elementów Skonczonych

Abstract The main idea of this work is to demonstrate an application of the generalized perturbation-based Stochastic Finite Element Method for a determination of the reliability indicators concerning elastic stability for a certain spectrum of the civil engineering structures. The reliability indicator is provided after the Eurocode according to the First Order Reliability Method, and computed using the higher order Taylor expansions with random coefficients. Computational implementation provided by the hybrid usage of the FEM system ROBOT and the computer algebra system MAPLE enables for reliability analysis of the critical forces in the most popular civil engineering structures like simple Euler beam, 2 and 3D single and multi-span steel frames, as well as polyethylene underground cylindrical shell. A contrast of the perturbation-based numerical approach with the Monte-Carlo simulation technique for the entire variability of the input random dispersion included into the Euler problem demonstrates the probabilistic efficiency of the perturbation method proposed.

Stochastic Finite Element Method for Nonlinear Dynamic Problem with Random Parameters

Several algorithms were proposed relating to the development of a framework of the perturbation-based stochastic finite element method (PSFEM) for nonlinear dynamic problem with random parameters, for this purpose, based on the stochastic virtual work principle , some algorithms and a framework related to SFEM have been studied. An interpolation method was used to discretize the random fields, which is based on representing the random field in terms of an interpolation rule involving a set of deterministic shape functions. Direct integration Wilson- Method in conjunction with Newton-Raphson scheme was adopted to solve finite element equations. Numerical examples were compared with Monte-Carlo simulation method to show that the approaches proposed herein are accurate and effective for the nonlinear dynamic analysis of structures with random parameters

Study on high order perturbation-based nonlinear stochastic finite element method for dynamic problems

Several algorithms were proposed relating to the development of a framework of the perturbation-based stochastic finite element method (PSFEM) for large variation nonlinear dynamic problems. For this purpose, algorithms and a framework related to SFEM based on the stochastic virtual work principle were studied. To prove the validity and practicality of the algorithms and framework, numerical examples for nonlinear dynamic problems with large variations were calculated and compared with the Monte-Carlo Simulation method. This comparison shows that the proposed approaches are accurate and effective for the nonlinear dynamic analysis of structures with random parameters.

Dynamic Analysis of Frame Structure Using Nonlinear Stochastic Finite Element Method

The objective of this paper is to propose an algorithms and a frame work of perturbation-based stochastic finite element method for large variation nonlinear dynamic problem of ship frame structure. For this purpose, based on the principle of stochastic virtual work some algorithms and a framework related to stochastic finite element method have been studied. To prove the validity and practicality, a numerical example for nonlinear dynamic problem with large variation was calculated to compare with Monte-Carlo simulation method. The results showed that the approaches proposed herein are feasible and effective for the nonlinear dynamic analysis of structures with random parameters.

Robust Optimization of Thermal-Dynamic Coupling Systems Using a Kriging Model

Thermally induced vibration of theflexible appendages of large-scale space structuresmay cause adverse effects to astronautical activities. This kind of unexpected response could be controlled to an admissible level through some optimal design methods. Because uncertainties in material properties and structural dimensions are inevitable in practice, they should also be considered during the optimization process to ensure the robust performance of the resultant structure. However, traditional robust design schemes based on either a direct Monte Carlo method or a perturbation-based stochastic finite elementmethod are usually too time-consuming to accomplish this task, because highly nonlinear analysis needs to be repeatedly conducted for this transient problem during the optimization process. To solve this problem, a robust optimization scheme based on the kriging model is developed in this paper. Because of the kriging approximation, only a few thermal–dynamic coupling analyses are needed to construct a satisfactory approximation function of the structural response. Based on this function, the mean value and the standard deviation of the structural response can be obtained efficiently, which greatly facilitates the calculation of the Pareto front for this kind of robust optimal design.

Finite element analysis with uncertain probabilities

This paper presents a general method for the finite element analysis of linear mechanical systems by taking into account probability density functions whose parameters are affected by fuzziness. Within this framework, the standard perturbation-based stochastic finite element method is relaxed in order to incorporate uncertain probabilities in static, dynamic and modal analyses. General formulae are provided for assessing the (fuzzy) structural reliability and several typologies of optimization problems (reliability-based design, robust design, robust/reliability-based design) are formalized. In doing this the credibility theory is extensively used to extract qualified crisp data from the available set of fuzzy results, so that standard optimizers can be adopted to solve the most important design problems. It is shown that the proposed methodology is a general and versatile tool for finite element analyses because it is able to consider, both, probabilistic and non-probabilistic sources of uncertainties, such as randomness, vagueness, ambiguity and imprecision.

Potential problems with random parameters by the generalized perturbation-based stochastic finite element method

The main aim of this paper is to present the new version and the application of the generalized nth order stochastic perturbation technique to the solution of the field problems with random coefficients. The variational stochastic nth order principles are proposed here using Taylor expansions with random parameters together with the additional stochastic finite element method discretization of the problem; a derivation of up to the fourth probabilistic moments and the coefficients of the potential function are also provided. The formulation is dual in the sense that both Direct Differentiation Method and the Response Function Method in its local version are proposed and implemented using the symbolic computer system MAPLE to determine numerically the random system response and to make the additional visualization. This methodology is illustrated with two examples - 1D viscous irrotational and incompressible fluid flow and 2D torsion of the prismatic rectangular beam with random transverse modulus.

Effects of random material and geometrical properties on structural safety of steel-concrete composite systems

A stochastic finite element-based algorithm for probabilistic analysis of structural systems made of composite sections with random material and geometrical properties under earthquake forces is proposed in this paper. Uncertainties in the structural parameters can be taken into account in this algorithm. For the perturbation-based stochastic finite element method, only the first two moments of random variables need to be known, and is numerically much more efficient and feasible than simulation techniques. The efficiency and accuracy of the proposed algorithm are validated by comparison with results of Monte Carlo simulation method. A summary of stiffness matrix formulation and perturbation-based stochastic finite element dynamic analysis formulation of structural systems made of composite sections is given. These are followed by suitable numerical examples, which indicate that employment of such a dynamic stochastic finite element method leads to significant economical, efficient and accurate solutions for the dynamic analysis of composite structures with stochastic parameters under earthquake forces. Copyright © 2010 John Wiley & Sons, Ltd.

A sub-domain method for solving stochastic problems with large uncertainties and repeated eigenvalues

The traditional perturbation-based stochastic finite element method has the inherent limitation of requiring small uncertainties to ensure reliable results. To overcome this limitation, this paper presents a new method that redefines the stochastic problem into some sub-domains of random variables. Then, the stochastic structural responses defined in the global domain can be explicitly reconstructed from the responses obtained in each sub-domain. In this way, the uncertainties of random variables could be relatively large to achieve the acceptable accuracy. In addition, this method is also able to deal with stochastic problems with repeated eigenvalues. Numerical examples are presented to illustrate the validity and capability of this new method by comparing the results with ones from Monte Carlo simulations. Copyright © 2009 John Wiley & Sons, Ltd.

Perturbation-based stochastic finite element method using polynomial response function for the elastic beams

The main goal here is to show the implementation of the perturbation-based stochastic finite element method using the polynomial response function in static analysis of the elastic beams. The polynomial response functions reconstruction is based on a solution of the several deterministic problems in each nodal point of the mesh, where random parameter is modified each time accordingly around its mean value. The polynomial functions relating the global displacements vector components and this input parameter are quite efficient in further determination of the partial derivatives of the state functions with respect to this parameter and of probabilistic moments is relatively easy.

On stochastic finite element method for linear elastostatics by the Taylor expansion

The main aim of the paper is to present an application of the Taylor expansion in formulation and computational implementation of the perturbation-based stochastic finite element method. Random-input parameters as well as all-state functions included in static equilibrium equations are expanded in this approach around their expectations via Taylor series up the order given a priori. It further enables a dual computational approach for determination of probabilistic moments of the state functions—a formation and the solution of increasing order equilibrium equations and, on the other hand, polynomial approximation of deterministic state functions with respect to a given input random parameter. Theoretical and technical details of such methodology are explained also; some elementary engineering application with analytical solution is available to derive explicitly fundamental probabilistic moments of the resulting state function.

Generalized perturbation-based stochastic finite element method in elastostatics

Generalised nth order stochastic perturbation technique that can be applied to solve some boundary value or boundary initial problems in computational physics and/or engineering with random coefficients is presented here. This technique is implemented in conjunction with the finite element method (FEM) to model 1D linear elastostatics problem with a single random variable. Main motivation of this work is to improve essentially the accuracy of the stochastic perturbation technique, which in its second order realization was ineffective for large variations of the input random fields. The nth order approach makes it possible to specify the accuracy of the computations a priori for the expected values and variances separately. The symbolic computer program is employed to perform computational studies on convergence of the first two probabilistic moments for simple unidirectional tension of the bar. These numerical studies verify the influence of coefficient of variation of the random input and, in the same time, of the perturbation parameter on the first four probabilistic moments of the final solution vector.

Design change of frame structure to enhance structural reliability

A method of design change aimed at the enhancement of structural reliability is formulated. The perturbation-based stochastic finite element method is employed to evaluate the reliability index of the advanced first-order second-moment method. The enhancement of the reliability index is devised by means of the minimal design change from a baseline design, while the shift of the reliability index is incorporated in terms of an equality constraint condition. The validity of the proposed method is verified by the numerical example of a two-storeyed portal frame with uncertain Young's modulus, for which the moment of inertia of the frame members is taken as design variables.