Spectral Stochastic Finite Element Research Articles

A domain decomposition solver for spectral stochastic finite element: an approximate sparse expansion approach

Spectral stochastic finite element (SSFE) has been widely used in the uncertainty quantification of real-life problems. However, the prohibitive computational burden prevents the application of the method in practical engineering systems because an enormous augmented system has to be solved. Although the domain decomposition method has been introduced to SSFE to improve the efficiency for the solution of the augmented system, there still exist significant challenges in solving the extended Schur complement (e-SC) system from domain decomposition method. In this paper, we develop an approximate sparse expansion-based domain decomposition solver to generalize the application of SSFE. An approximate sparse expansion is first presented for the subdomain-level augmented matrix so that the computational cost in each iteration of the preconditioned conjugate gradient is greatly alleviated. Based on the developed sparse expansion, we further establish an approximate sparse preconditioner to accelerate the convergence of the preconditioned conjugate gradient. The developed approximate sparse expansion-based domain decomposition solver is then incorporated in the context of SSFE. Since the difficulties of solving the e-SC system have been overcome, the developed approximate sparse expansion-based solver greatly improves the computational efficiency of the solution of the e-SC system, and thereby, the SSFE is capable of dealing with large-scale engineering systems. Two numerical examples demonstrate that the developed method can significantly enhance the efficiency for the stochastic response analysis of practical engineering systems.

Exploitation of the Spectral Stochastic Finite Element Model for the Evaluation of Surface Defects of the CFRP Composite

Exploitation of the Spectral Stochastic Finite Element Model for the Evaluation of Surface Defects of the CFRP Composite

Modeling of spatial permittivity variations using Karhunen–Loève expansion for stochastic electromagnetic problems

Fabrication tolerance often causes spatial variations in material properties in electromagnetic systems. In this paper, these spatial variations are modeled using Karhunen–Loéve expansion, which is truncated for a finite dimensional representation. Uncertainty quantification is performed by incorporating this model into an intrusive polynomial chaos expansion based spectral stochastic finite element method. Using a numerical example, it has been shown that the stochastic response obtained by the proposed approach is accurate and computationally efficient. The impact of correlation length of the covariance function used in the spectral expansion is analyzed.

Applying the spectral stochastic finite element method in multiple-random field RC structures

Applying the spectral stochastic finite element method in multiple-random field RC structures

A Novel Method for Intrusive Stochastic Estimation of Geometric Tolerance Effects in Finite Element Electromagnetic Analysis

The robust design of microwave, millimeter-wave, or Terahertz components and structures incorporating manufacturing process tolerances would enhance the fabrication yield and thereby reduce the overall production cost. Based on the spectral stochastic finite element method (SSFEM) for material variations, a novel geometrical SSFEM (GSSFEM) is proposed to analyze dimensional uncertainty in 3-D microwave models. The Jacobian for Piola mapping is represented as a function of stochastic variables to capture mesh level uncertainties. Polynomial chaos expansion is used to approximate the electric field as a random process. The technique is validated by applying it to waveguide problems with uncertainty in geometric dimensions. GSSFEM results are compared with analytical formulations of the transmission coefficient for a physical insight and further with the Monte Carlo simulations for quantitative evaluation. The results using the proposed approach are in excellent agreement with conventional methods and are found to be faster than sparse grid stochastic collocation, while the scaling of the computational requirements with the number of degrees of freedom and stochastic dimension is as good as this efficient scheme. Furthermore, this approach has been employed to analyze uncertainty in multiple geometric parameters to compare their sensitivity. The proposed GSSFEM can be employed for analyzing the impact of fabrication tolerance of passive waveguide components at microwave frequencies and beyond.

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.

Simplified spectral stochastic finite element formulations for uncertainty quantification of engineering structures

In general, the structure inputs including loading and physical specifications can be deterministic or stochastic and uncertainties can be modeled using a random variable, random field, or a stochastic process. A well-known method developed to analyze structures with uncertainties is the spectral stochastic finite element method (SSFEM). To improve the efficiency of SSFEM, this paper has developed three simplified formulations to separately solve problems involving stochastic loading and constant material properties, stochastic loading and random material properties and stochastic loading and stochastic material properties the main feature of which is that they need fewer finite element (FE) analyses to assess the statistical specifications of the response. To examine their accuracy and efficiency, three practical engineering problems: 1) Euler-Bernoulli beam, 2) square solid and 3) modeling the corrosion effect in a concrete structure were solved and the results were compared with those of the Monte Carlo simulation (MCS). To see if the proposed formulations can be used in structural reliability analyses, a limit state function (LSF) was defined for each problem and the results were compared with those of the corresponding MCS; their significant efficiency and enough potential was, thus, confirmed.

Uncertainty analysis of higher-order sandwich beam using a hybrid stochastic time-domain spectral element method

A stochastic time-domain spectral element method (STSEM) is applied to sandwich beam structures. STSEM couples the computational efficiencies of the spectral stochastic finite element method and the time-domain spectral element method. A higher-order sandwich panel theory incorporating the core flexibility is considered for the analysis. The material properties are modeled as random fields and represented by the Karhunen-Loéve expansion. The stochastic nodal displacements are expressed using the polynomial chaos expansion, and the expansion coefficients are computed by solving the spectral stochastic finite element equations. The computational efficiency of the proposed method is compared with Monte-Carlo simulation.

Stochastic strain and stress computation of a higher-order sandwich beam using hybrid stochastic time domain spectral element method

In this work, a combination of the spectral stochastic finite element method (SSFEM) and the time-domain spectral element method (TSEM), referred to as the stochastic time domain spectral element method (STSEM), is presented to compute the stochasticity of strain and stress of a higher-order sandwich composite beam with spatial variability in the material properties. The method proposed in this work employs the efficiencies of both SSFEM and TSEM for the uncertainty analysis of a sandwich beam. The material properties of face sheets and core are considered as Gaussian random fields, which are discretized using the Karhunen-Loéve expansion, and polynomial chaos expansion is used to represent the response quantity. A numerical example is considered for which, first, a sensitivity analysis is performed to identify the most sensitive material properties. Then, the proposed STSEM is used to demonstrate the computational efficiency and numerical accuracy in comparison with Monte-Carlo simulation. Moreover, the effect of core depth on strain and stress variability is also examined.

Hybrid uncertainty analysis of functionally graded plates via multiple-imprecise-random-field modelling of uncertain material properties

This paper presents the mixed probabilistic/non-probabilistic uncertainty problem regarding the hybrid uncertainty propagation on physical model of functionally graded plates through dissimilarly uncertain characterisation of material properties in the presence of the multiple-imprecise-random-field modelling. The spectral stochastic finite element framework is extended to perform mixed uncertainty analysis of functionally graded material whereby the spatial variation of elastic modulus of the plate is uniquely characterised as the multiple-imprecise-random-field modelling more than a purely homogeneous random field. This characterisation has originated in heeding the construction of physical models and the significant influences of separately uncertain characteristics propagated on the physical model through different material ingredients are examined to make physical models more reliable and realistic. Mechanics and mathematics intrusive-formulations are then established to explicitly produce the uncertain-but-bounded statistical characteristics of responses, namely interval mean value, interval standard deviation. Bounded surfaces of statistical moments and bounding distribution functions are vividly depicted for demonstration. The feasibility and effectiveness of the presented work are illustrated via test cases of numerical examples.

Stochastic Modeling and Reliability Analysis of Wing Flutter

In this work, a physics-based first-order reliability method (FORM) algorithm is proposed for the flutter reliability analysis of an aircraft wing in the frequency domain. The limit state function, which is an implicit function of random variables, is defined in terms of the damping ratio of the aeroelastic system in a conditional sense on flow velocity. Two aeroelastic cases, namely, an airfoil section model and a cantilever wing model, are considered for carrying out the studies. These aeroelastic models have well separated mean bending and mean torsional modal frequencies. The geometric, structural, and aerodynamic parameters of airfoil and wing systems are modeled as independent Gaussian random variables. The effects of these on the statistics of frequency and damping ratio, and the cumulative distribution functions (CDFs) of flutter velocity are studied. In the case of the wing, the effects of modeling stiffness parameters as Gaussian random fields on the CDFs of flutter velocity are also studied. Here, spectral stochastic finite element method (SFEM) based on Karhunen–Loeve (K–L) expansion is used to discretize the random fields into random variables. From the study of an airfoil system, it is observed that parameters like torsional stiffness, elastic axis location, free stream density, and mass moment of inertia are more sensitive as compared with other parameters. However, in the case of the wing parameters such as torsional stiffness, free stream density, mass moment of inertia, and mass are observed to be more sensitive. The CDFs of flutter velocity obtained using the proposed algorithm are compared with Monte Carlo simulations (MCS) and found to be accurate. A comparative study of aeroelastic reliability for the wing is also carried out by treating stiffness parameters as random variables and random fields. It is observed that the CDFs of flutter velocity in the tail region are conservative when stiffness parameters are treated as random variables.

A natural frequency sensitivity-based stabilization in spectral stochasticfinite element method for frequency response analysis

In applying the spectral stochastic finite element methods to the frequency response analysis, the conventional methods are known to give unstable and inaccurate results near the natural frequencies. To address this issue, a new sensitivity based stabilized formulation for stochastic frequency response analysis is proposed in this paper. The main difference over the conventional spectral methods is that the polynomials of random variables are applied to both numerator and denominator in approximating the harmonic response solution. In order to reflect the resonance behavior of the structure, the denominator polynomials is constructed by utilizing the natural frequency sensitivity and the random mode superposition. The numerator is approximated by applying a polynomial chaos expansion, and its coefficients are obtained through the Galerkin or the spectral projection method. Through various numerical studies, it is seen that the proposed method improves accuracy, especially in the vicinities of structural natural frequencies compared to conventional spectral methods.

Digital-Image-Driven Stochastic Homogenization for Recycled Aggregate Concrete Based on Material Microstructure

In this paper, a digital-image-driven stochastic homogenization method is developed to analyze elastic heterogeneous media such as recycled aggregate concrete (RAC), etc. This linking can be accomplished in an efficient manner by exploiting the excellent synergy of finite pixel-element method and Monte Carlo simulation for the computation of the effective properties of the random five-phase composite. The pixel-point discretization of system geometry is used for the approximation of the mechanical response of the elastic heterogeneous microstructure. Using nanoindentation technique and scanning electron microscopy, tens of digital images of modulus map of the five-phase heterogeneous material are made. Using the moving window technique and the Monte Carlo method, the random elastic moduli of the five phases at micro-scale are obtained. Then the effective elastic modulus of the meso-scale representative volume element (RVE) is computed based on spectral stochastic finite element method. Finally, the effective modulus is used to analyze the global behavior of RVE at macro-scale. Then the finite pixel-element method is used to investigate the effect of microscopic covariance noise on the global material properties, as well as the computational efficiency. The results show that the digital image method is an accurate and efficient tool to investigate the random material properties across scales.

Symplectic Approach on the Wave Propagation Problem for Periodic Structures with Uncertainty

An effective method is proposed in this paper for the symplectic eigenvalue problem of a periodic structure with uniform stochastic properties. Since structural parameters are uncertain, the symplectic transfer matrix of a typical substructure, which is described in the state space, also has stochastic properties. An effective spectral stochastic finite element method is adopted to ensure the symplectic orthogonality of the random symplectic matrix on the premise of certain precision. By means of the Rayleigh quotient method, the symplectic eigenvalue problem of random symplectic matrix is investigated. On the basis of this, the mean value and the standard deviation of the random eigenvalues are fully discussed. The comparison between the numerical results derived from the proposed method and the Monte-Carlo simulation indicates that the proposed method has high precision. This research provides a useful guidance for the dynamic analysis of periodic structures with stochastic properties.

Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

The implementation of spectral stochastic Galerkin finite element approximation methods for Helmholtz equations with random inputs is considered. The corresponding linear systems have matrices represented as Kronecker products. The sparsity of such matrices is analysed and a mean-based preconditioner is constructed. Numerical examples show the efficiency of the mean-based preconditioners for stochastic Helmholtz problems, which are not too close to a resonant frequency.

A spectral stochastic formulation for nonlinear framed structures

The present paper proposes a stochastic formulation which enables the effective coupling of spectral stochastic finite elements with geometrically nonlinear analysis of framed structures. This is achieved by projecting the stochastic part of the incremental displacements, formulated in the framework of a Newton–Raphson solution scheme, to a polynomial chaos expansion basis. The nonlinear solution is consequently achieved by introducing these expansion coefficients in the iterative solution algorithm as unknowns to be evaluated. With this approach, an augmented system of nonlinear equations is produced and the values of the coefficients are obtained through convergence of the iterative algorithm. In a similar to the deterministic problems manner, the nonlinear load or displacement control algorithms are extended to their stochastic counterparts and a discussion on the appropriate choice of such algorithms depending on the problem at hand is presented. This work focuses on stochastic beam finite element systems in which uncertainty is considered in the system properties. The efficiency of the proposed methodology is demonstrated in benchmark problems with strong geometrically nonlinear behavior. In order to verify the validity of the proposed approach the results obtained are compared to those of Monte Carlo simulations and fair accuracy can be reported.

Cost reduction of stochastic Galerkin method by adaptive identification of significant polynomial chaos bases for elliptic equations

One widely used and computationally efficient method for uncertainty quantification using spectral stochastic finite element is the stochastic Galerkin method. Here the solution is represented in polynomial chaos expansion, and the residual of the discretized governing equation is projected on the polynomial chaos bases. This results in a system of deterministic algebraic equations with the polynomials chaos coefficients as unknown. However, one impediment for its large scale applications is the curse of dimensionality, that is, the exponential growth of the number of polynomial chaos bases with the stochastic dimensionality and degree of expansion. Here, for a stochastic elliptic problem, an adaptive selection of polynomial chaos bases is proposed. Accordingly, during the first few iterations in the preconditioned conjugate gradient method for solving the system of linear algebraic equations, the chaos bases with maximal contribution –in an appropriately defined metric – to the solution are first identified. Subsequently, only these bases are retained for further iterations until convergence is achieved. Using numerical studies a three times cost saving over the existing method is observed. Furthermore, for enhancing the computational cost gain, the stochastic Galerkin method is reformulated as a generalized Sylvester equation. This step allowed efficient usage of the sparsity of moments of product of polynomial chaos bases. Through numerical studies on problems with large stochastic dimensionality, an additional cost saving of up to one order of magnitude –twenty times –is observed. This amounts to sixty times speedup over the existing method, when adaptive selection and generalized Sylvester equation formulation are used together. The proposed methodology can be easily incorporated in an existing standard stochastic Galerkin method solver for elliptic problems.

Computational structural analysis of composites with spectral-based stochastic multi-scale method

Composite materials and structures may be characterized at different length scales, ranging from the micro-scale at the fiber–matrix level, meso-scale at the lamina–laminate level, to structural macro-scale. However, uncertainties in material properties and geometric parameters due to manufacturing, defects and assembly processes may occur at various length scales. This paper presents a computational framework for stochastic analysis of composites with consideration of stochastic parameters at micro- and meso-scales. The novelty of the proposed framework is the integration of the spectral stochastic finite element method and asymptotic homogenization method within a finite element technique, which was implemented through $$\hbox {ABAQUS}^{\circledR }$$ . This spectral stochastic homogenization method efficiently predicts the propagation of uncertainties from the constituent to ply levels. The derived probability distributions of effective properties were verified by Monte Carlo simulation. Another novelty is the study of the influence of stochastic parameters at both micro-scale and meso-scale on the failure prediction of composite structures, without assumptions of probabilistic characteristic of ply properties commonly used in a single-scale stochastic analysis. The up-scaled uncertainties combined with other randomness at meso-scale (strength properties and ply orientations) are provided as the input of meso-scale stochastic strength analysis of a quasi-isotropic laminate based on classical lamination theory (CLT) and ply discount. The probability distribution of first-ply failure and ultimate failure loads are obtained and their sensitivity factors with respect to input variations are presented.

A Stochastic Dual Response Surface Method for Reliability Analysis Considering the Spatial Variability

To solve spectral stochastic finite element problems, the collocation-based spectral stochastic finite element method (SSFEM) was developed, and the Stochastic Response Surface Method (SRSM) was used to represent uncertainty propagation. Therefore, the accuracy of SRSM is important for obtaining more accurate probabilistic results. The weighted SRSM was developed to improve the global accuracy of SRSM, but it is not suitable for random field problems because weights might distort the response surface. In this study, a new Stochastic Dual-response Surface Method (SDRSM) was developed to improve the accuracy of SRSM. The SDRSM combines the conventional SRSM and target-weighted SRSM (TWSRSM), which is assigned a weight for the numerical result corresponding to the collocation point. Then, the proposed method was extended to deal with problems involving random fields. For comparison with the conventional methods (SRSM and WSRSM), two numerical examples involving random fields were carried out. Compared with Monte Carlo simulation results, SDRSM shows the smallest error without the addition of the collocation points. In addition, the mean absolute errors for equally spaced probability intervals were compared, and their mean and standard deviation of SDRSM were relatively smaller than that of other methods.

Inexact Methods for Symmetric Stochastic Eigenvalue Problems

We study two inexact methods for solutions of random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric matrix operator, the methods solve for eigenvalues and eigenvectors represented using polynomial chaos expansions. Both methods are based on the stochastic Galerkin formulation of the eigenvalue problem and they exploit its Kronecker-product structure. The first method is an inexact variant of the stochastic inverse subspace iteration [B. Soused\'{\i}k, H. C. Elman, SIAM/ASA Journal on Uncertainty Quantification 4(1), pp. 163--189, 2016]. The second method is based on an inexact variant of Newton iteration. In both cases, the problems are formulated so that the associated stochastic Galerkin matrices are symmetric, and the corresponding linear problems are solved using preconditioned Krylov subspace methods with several novel hierarchical preconditioners. The accuracy of the methods is compared with that of Monte Carlo and stochastic collocation, and the effectiveness of the methods is illustrated by numerical experiments.