Stochastic Finite Element Analysis Research Articles

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.

A novel joint diagonalization approach for linear stochastic systems and reliability analysis

PurposeThe purpose of this paper is to present a novel gradient‐based iterative algorithm for the joint diagonalization of a set of real symmetric matrices. The approximate joint diagonalization of a set of matrices is an important tool for solving stochastic linear equations. As an application, reliability analysis of structures by using the stochastic finite element analysis based on the joint diagonalization approach is also introduced in this paper, and it provides useful references to practical engineers.Design/methodology/approachBy starting with a least squares (LS) criterion, the authors obtain a classical nonlinear cost‐function and transfer the joint diagonalization problem into a least squares like minimization problem. A gradient method for minimizing such a cost function is derived and tested against other techniques in engineering applications.FindingsA novel approach is presented for joint diagonalization for a set of real symmetric matrices. The new algorithm works on the numerical gradient base, and solves the problem with iterations. Demonstrated by examples, the new algorithm shows the merits of simplicity, effectiveness, and computational efficiency.Originality/valueA novel algorithm for joint diagonalization of real symmetric matrices is presented in this paper. The new algorithm is based on the least squares criterion, and it iteratively searches for the optimal transformation matrix based on the gradient of the cost function, which can be computed in a closed form. Numerical examples show that the new algorithm is efficient and robust. The new algorithm is applied in conjunction with stochastic finite element methods, and very promising results are observed which match very well with the Monte Carlo method, but with higher computational efficiency. The new method is also tested in the context of structural reliability analysis. The reliability index obtained with the joint diagonalization approach is compared with the conventional Hasofer Lind algorithm, and again good agreement is achieved.

Probabilistic optimal safety with minimum life-cycle cost based on stochastic finite element analysis of steel cable-stayed bridges

This study was intended to efficiently perform the probabilistic safety and optimal design assessment of steel cable-stayed bridges (SCS bridges) using stochastic finite element analysis (SFEA) and expected life-cycle cost (LCC) concept. To that end, advanced probabilistic finite element algorithm (APFEA) which enables to execute the static and dynamic SFEA considering aleatory uncertainties contained in random variable was developed. APFEA is the useful analytical means enabling to conduct the reliability assessment (RA) in a systematic way by considering the result of SFEA based on linearity and nonlinearity of before or after introducing initial tensile force. Appropriateness of APFEA was verified in such a way of comparing the result of SFEA of a simple structure and the result of numerical analysis using Monte Carlo Simulation (MCS) program. The probabilistic method of SCS bridges was set, taking into account of analytical parameters. The dynamic response characteristic by probabilistic method was evaluated using ASFEA, and RA was carried out based on analysis result, thereby quantitatively calculating the probabilistic safety. The optimal design of SCS bridges was determined based on the expected LCC according to the results of SFEA and RA of alternative designs. Moreover, given the potential epistemic uncertainty contained in safety index, failure probability and minimum LCC, the sensitivity analysis was conducted and as a result, a critical distribution phase was illustrated using a cumulative-percentile.

Reliability assessment of 3D space frame structures applying stochastic finite element analysis

Throughout design development of satellite structure, stress engineer is usually challenged with randomness in applied loads and material properties. To overcome such problem, a risk-based design is applied which estimates satellite structure probability of failure under static and thermal loads. Determining probability of failure can help to update initially applied factors of safety that were used during structure preliminary design phase. These factors of safety are related to the satellite mission objective. Sensitivity-based analysis is to be implemented in the context of finite element analysis (probabilistic finite element method or stochastic finite element method (SFEM)) to determine the probability of failure for satellite structure or one of its components.

Structural Optimization Based on Stochastic Finite Element Analysis and Optimality Criteria Methods

An approach for structural optimization with uncertainty is proposed in this paper, which is based on stochastic finite element analysis and an optimality criteria method. The sensitivity analysis of the structural failure probability via design variables are performed by difference method according to stochastic finite element analysis. The information of the partial derivatives is incorporated into the optimality criteria method. The optimization of the section of a beam with uncertainty, as a numerical example, is illustrated by the approach. The results demonstrate the efficiency of the approach.

Dynamic Optimization Design of Stochastic Finite Element

A general model of the optimum design of structural vibration with the random parameters is introduced. Dynamic optimization design of stochastic finite element using the Preconditioned Conjugate Gradient method (PCG) is presented. The vibration equation of a system is transformed to a static problem by using the Willson method and stochastic finite element analysis of structural vibration adopts the PCG.An example is given respectively and the proposed method is robust and efficient.

Response variability of laminate composite plates due to spatially random material parameter

Young’s modulus is at the center of attention in the stochastic finite element analysis since the parameter plays an important role in determining structural behavior. However, the other material parameter of Poisson’s ratio is another independent material parameter that governs the behavior of structural systems. Accordingly, the independent estimation of the influence of this parameter on the uncertain response of a system is of importance from the perspective of stochastic analysis. To this end, we propose a formulation to determine the response variability in laminated composite plates due to the spatial randomness of Poisson’s ratio. To filter out the independent contribution of random Poisson’s ratio, a decomposition of the constitutive matrix into several sub-matrices by using the Taylor’s expansion is needed, which makes the random Poisson’s ratio simple enough to be included in the formulation. To validate the adequacy of the proposed formulation, several examples are chosen and the results are compared with those given by Monte Carlo analysis. By means of the formulation suggested here, it is expected that an extension of the formulation to include the effect of correlations between random Poisson’s ratio and other structural and/or geometrical parameters will be achieved with ease, resulting in a more practical estimation of the response variability of laminated composite plates.

Stochastic finite element analysis of composite plates considering spatial randomness of material properties and their correlations

Considering the randomness of material parameters in the laminated composite plate, a scheme of stochastic finite element method to analyze the displacement response variability is suggested. In the formulation we adopted the concept of the weighted integral where the random variable is defined as integration of stochastic field function multiplied by a deterministic function over a finite element. In general the elastic modulus of composite materials has distinct value along an individual axis. Accordingly, we need to assume 5 material parameters as random. The correlations between these random parameters are modeled by means of correlation functions, and the degree of correlation is defined in terms of correlation coefficients. For the verification of the proposed scheme, we employ an independent analysis of Monte Carlo simulation with which statistical results can be obtained. Comparison is made between the proposed scheme and Monte Carlo simulation.

Efficient stochastic structural analysis using Guyan reduction

This paper introduces the application of the Guyan reduction within the stochastic finite element (SFE) analysis, which employs a Galerkin-based Polynomial chaos (P-C) expansion formulation. It is shown that by reducing the size of the deterministic FE model, a substantial improvement in the overall computational efficiency can be achieved. An implementation exploiting the features of the proposed formulation is presented. In this regard, especially the interaction with the 3rd party FE solvers has been addressed. The suggested method has been tested on a simple grid structure and also on a large building model, where the accuracy and efficiency of the introduced approach have been quantified.

A reduced spectral function approach for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations with non-Gaussian random fields are considered. A novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated. It is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors. These functions, called as the spectral functions, can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation. Based on the projection in a reduced orthonormal vector basis, a Galerkin error minimization approach is proposed. The constants appearing in the Galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation. A hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution. The method is illustrated using the stochastic nanomechanics of a zinc oxide (ZnO) nanowire deflected under the atomic force microscope (AFM) tip. The results are compared with the results obtained using direct Monte Carlo simulation, classical Neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness.

On the Capabilities of the Polynomial Chaos Expansion Method within SFE Analysis—An Overview

This paper addresses the most recent developments concerning the application of the P-C expansion method within the Stochastic Finite Element (SFE) analysis, in particular considering computational solid mechanics. More specifically, the focus has been on the use of the method for the propagation of the stochastic structural responses due to the extensive amount of contributions in this context. Numerical examples presented in the literature are listed in this regard, in order to shed some light on the range of applications, especially in terms of the probabilistic dimensionality of the problem. Furthermore, the recently emerging utilization of the method within the modeling of uncertain input parameters is covered briefly. With this contribution, it is aimed to present an overview on the state-of-art of the P-C literature and the capabilities of the method.

A study on safety assessment of cable-stayed bridges based on stochastic finite element analysis and reliability analysis

It is more reasonable to evaluate the effect of uncertainty included in actual structures using the reliability assessment method. However, using the reliability assessment method to evaluate complicated structures, such as cable-stayed bridges whose limit state function is given as a nonlinear function, was very difficult. Therefore, in this study the structural response characteristics are investigated quantitatively using Stochastic Finite Element Analysis (SFEA) according to the external uncertainty of random variables included in cable-stayed bridges, and Reliability Analysis (RA) considering external uncertainty is performed more efficiently using the result of SFEA. For this approach, after performing the initial shape analysis of the cable-stayed bridge, improved Stochastic Finite Element Method Program (SFEMP) is developed for executing nonlinear SFEA by considering correlation of random variables. The feasibility of the developed program was investigated by comparing the analysis results of beam-cable structures with those of the verified program using Direct Monte Carlo Simulation (DMCS). The effect of each random variable on the structural response was examined by performing sensitivity analysis. The safety index and failure probability according to the correlation was investigated by applying RA that considered the correlation of the major random variables. The critical percentile distribution of the RA result is re-evaluated quantitatively considering the effect of the internal uncertainty.

Adaptive sparse polynomial chaos expansion based on least angle regression

Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e. of Galerkin type) or non intrusive) unaffordable when the deterministic finite element model is expensive to evaluate. To address such problems, the paper describes a non intrusive method that builds a sparse PC expansion. First, an original strategy for truncating the PC expansions, based on hyperbolic index sets, is proposed. Then an adaptive algorithm based on least angle regression (LAR) is devised for automatically detecting the significant coefficients of the PC expansion. Beside the sparsity of the basis, the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon. The accuracy of the PC metamodel is checked using an estimate inspired by statistical learning theory, namely the corrected leave-one-out error. As a consequence, a rather small number of PC terms are eventually retained ( sparse representation), which may be obtained at a reduced computational cost compared to the classical “full” PC approximation. The convergence of the algorithm is shown on an analytical function. Then the method is illustrated on three stochastic finite element problems. The first model features 10 input random variables, whereas the two others involve an input random field, which is discretized into 38 and 30 − 500 random variables, respectively.

Stochastic Finite Element Analysis for Shrinkage and Creep of a Concrete Bridge Based on LHS

Shrinkage and creep are the inherent properties of concrete bridge. According to Latin Hypercube Sampling method (LHS) and concrete bridge shrinkage and creep uncertainty parameters, this paper promotes stochastic finite element method based on LHS for concrete bridge shrinkage and creep uncertainty analysis in ANSYS random analysis model. Based on this stochastic finite element method, internal force and long-term deformation of a continuous rigid frame bridge with density gradient concrete is analyzed. Result show, internal force and deformation of continuous rigid frame bridge after completing are reasonable and satisfy Code requirement, and are various in previous 3 years at service stage and trend to steadily after 3 years. Research result provides theoretical basis for shrinkage and creep uncertainty further analysis in concrete bridge.

Stochastic finite element analysis of long-span bridges with CFRP cables under earthquake ground motion

Stochastic seismic analysis of long-span bridges with Carbon fibre reinforced polymer (CFRP) cables are presented in this study through combination of the advantages of the perturbation based stochastic finite element method (SFEM) and Monte Carlo simulation (MCS) method. Jindo cable-stayed and Fatih Sultan Mehmet (Second Bosporus) suspension bridges are chosen as an example. Carbon fibre reinforced polymer cable (CFRP) and steel cables are used separately, in which the cable’s cross sectional area is determined by the principle equivalent axial stiffness. Geometric nonlinear effects are considered in the analysis. Uncertainties in the material are taken into account and Kocaeli earthquake in 1999 is chosen as a ground motion. The efficiency and accuracy of the proposed algorithm are validated by comparing with results of MCS method. It can be stated that using of CFRP cables in long-span bridges subjected to earthquake forces is feasible.

High dimensional model representation for stochastic finite element analysis

This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation.

Stochastic finite element analysis of a cable-stayed bridge system with varying material properties

Stochastic seismic finite element analysis of a cable-stayed bridge whose material properties are described by random fields is 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 structural system is given. The Jindo Bridge, constructed in South Korea, is chosen as a numerical example. The Kocaeli earthquake in 1999 is considered as a ground motion. During the stochastic analysis, displacements and internal forces of the considered bridge are obtained from perturbation based stochastic finite element method (SFEM) and MCS method by changing elastic modulus and mass density as random variable. The efficiency and accuracy of the proposed SFEM algorithm are evaluated by comparison with results of MCS method. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.

Stochastic finite element analysis of structural systems with partially restrained connections subjected to seismic loads

The present paper investigates the stochastic seismic responses of steel structure systems with Partially Restrained (PR) connections by using Perturbation based Stochastic Finite Element (PSFEM) method. A stiffness matrix formulation of steel systems with PR connections and PSFEM and MCS formulations of structural systems are given. Based on the formulations, a computer program in FORTRAN language has been developed, and stochastic seismic analyses of steel frame and bridge systems have been performed for different types of connections. The connection parameters, material and geometrical properties are assumed to be random variables in the analyses. The Kocaeli earthquake occurred in 1999 is considered as a ground motion. The connection parameters, material and geometrical properties are considered to be random variables. The efficiency and accuracy of the proposed SFEM algorithm are validated by comparison with results of Monte Carlo simulation (MCS) method.

Incomplete factorization preconditioners for the iterative solution of Stochastic Finite Element equations

This work is focused on enhancing the computational efficiency in Monte Carlo simulation-based Stochastic Finite Element (SFE) analysis of large-scale structural models. Such analyses require the solution of successive systems of equations derived during simulations, which can be efficiently treated using customized versions of the iterative Preconditioned Conjugate Gradient (PCG) solution method. PCG-customization is localized at the preconditioning matrix employed to accelerate convergence. Thus, specialized preconditioners following the rationale of incomplete factorization are presented, which retain only essential numerical information during factorization. The resulting PCG-based solution schemes allow for computationally efficient SFE analyses with low storage demands in computer memory.

An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis

Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e.of Galerkin type) or non-intrusive) unaffordable when the deterministic finite element model is expensive to evaluate. To address such problems, this paper describes a non-intrusive method that builds a sparse PC expansion. An adaptive regression-based algorithm is proposed for automatically detecting the significant coefficients of the PC expansion. Besides the sparsity of the basis, the experimental design used at each step of the algorithm is systematically complemented in order to ensure the well-posedness of the various regression problems. The accuracy of the PC model is checked using classical tools of statistical learning theory (e.g. leave-one-out cross-validation). As a consequence, a rather small number of PC terms is eventually retained (sparse representation), which may be obtained at a reduced computational cost compared to the classical “full” PC approximation. The convergence of the algorithm is shown on an academic example. Then the method is illustrated on two stochastic finite element problems, namely a truss and a frame structure involving 10 and 21 input random variables, respectively.