Spectral Stochastic Finite Element Method Research Articles

Seismic analysis of gravity dam-foundation systems using stochastic spectral finite element method

Seismic analysis of dam-foundation systems is inherently complex due to uncertainties in material properties. This study investigates the effects of material randomness and heterogeneity on the seismic response of gravity dam-foundation systems using the stochastic spectral finite element method (SSFEM) based on the Karhunen Loève expansion and the polynomial chaos expansion. The Pine Plane dam is taken as an analysis example. First, the appropriate Karhunen Loève expansion modes and polynomial chaos order are determined for SSFEM by comparing them with the Monte-Carlo simulations. Subsequently, the seismic response of the dam is analyzed in the stochastic and deterministic models, respectively. Results indicate that material randomness and heterogeneity may reduce the seismic response of gravity dams, and the deterministic model may overestimate dynamic responses. Subsequently, the parametric analysis of the correlation length, which measures the spatial correlation between variables in different positions and indicates the size of heterogeneity material patches, is conducted. It is demonstrated that the correlation length would not significantly affect the mean responses, but a larger correlation length leads to greater dispersions in responses. Finally, the effects of randomness in the foundation are investigated. The heterogeneities in the foundation affect the response of the dam in the high-frequency range. Therefore, the uncertainty in material properties should be fully considered in a structural assessment and risk analysis.

An Efficient SSFEM-POD Scheme for Wideband Stochastic Analysis of Permittivity Variations

Stochastic analysis using polynomial chaos expansion (PCE)-based spectral stochastic finite-element method (SSFEM) is computationally expensive and is impacted by the curse of dimensionality. In this article, a low-dimensional basis is developed to approximate the problem using proper orthogonal decomposition (POD) to employ the PCE-based SSFEM, thereby resulting in an efficient intrusive formulation for uncertainty quantification (UQ) over a wide frequency band. Galerkin’s projection of vector finite elements for an electromagnetic (EM) problem with relative permittivity variations in multiple dielectric regions is formulated. The proposed method is shown to be computationally efficient and gives accurate results for a range of frequencies with single computation. It is also shown that this approach does not scale adversely with the number of stochastic parameters and has significantly lower memory requirement.

New non-intrusive stochastic finite element method for plate structures

Currently, several intrusive stochastic finite element methods (SFEMs) such as perturbation method and spectral SFEM are widely applied for stochastic response analysis of continuous structures. However, the intrusive SFEMs need to modify conventional finite element formulations to establish the stochastic stiffness matrix, and cannot calculate the probability density function of structural response and the reliability straightforwardly. This paper proposes a new non-intrusive SFEM for efficiently computing stochastic responses and reliabilities of plates in a unified way. Firstly, the direct probability integral method (DPIM) is developed to obtain the probability density function of stochastic response by solving probability density integral equation (PDIE). Secondly, the non-intrusive SFEM based on DPIM decouples the deterministic finite element analysis and PDIE to calculate the stochastic responses and reliabilities of uncertain plate structures, and the discretization and quantification of random fields of elastic modulus and thickness are implemented through Karhunen-Loève expansion. Finally, several examples of uncertain Kirchhoff and Mindlin plates demonstrate the efficiency and versatility of the proposed non-intrusive method by comparing with the results from Monte Carlo simulation and literature. The effects of correlation length, mean and variability of random field on the probability distribution of responses and the reliabilities of plates are revealed. For Gaussian random thickness, the linearly elastic plate yields non-Gaussian distributed responses. Increasing the correlation length and variability of random field reduces the reliabilities of plates.

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.

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.

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.

A stochastic spectral finite element method for solution of faulting-induced wave propagation in materially random continua without explicitly modeled discontinuities

This paper proposes a new efficient stochastically adapted spectral finite element method to simulate fault dislocation and its wave propagation consequences. For this purpose, a dynamic form of the split node technique is formulated and developed to stochastic spectral finite element method in order to model fault dislocation happening within a random media without increasing computational demand caused by discontinuities. As discontinuities are not modeled explicitly herein, no additional degrees of freedom are implemented in the proposed method due to the discontinuities, while effects of these discontinuities are preserved. Therefore, the present method simultaneously includes merits of stochastic finite element method, spectral finite element method and the spilt node technique, thereby providing a new numerical tool for analysis of wave propagation under fault dislocation in random media. Several numerical simulations are solved by the proposed method, which present stochastic analysis of fault slip-induced wave propagation in layered random media. Formulations and numerical results demonstrate capability, application and efficiency of this novel method.

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.

Time integration of stochastic generalized equations of motion using SSFEM

The paper develops an integration approach to stochastic nonlinear partial differential equations (SPDE’s) with parameters to be random fields. The methodology is based upon assumption that random fields are from a special class of functions, and can be described as a product of two functions with dependent and independent random variables. Such an approach allows one to use Karhunen-Lo`eve expansion directly, and the modified stochastic spectral finite element method (SSFEM). It is assumed that a random field is stationary and Gaussian while the autocovariance function is known. A numerical example of onedimensional heat waves analysis is shown.

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 stochastic spectral finite element method for wave propagation analyses with medium uncertainties

• Wave propagation in a medium with stochastic properties is studied. • A stochastically enriched spectral finite element method (StSFEM) is applied. • StSFEM accelerates temporal integration schemes and decreases numerical dispersion. • It provides an efficient numerical solution for Fredholm equation of KL Expansion. • The StSFEM is developed to large-scale stochastic wave propagation phenomena. A stochastically enriched spectral finite element method (StSFEM) is developed to solve wave propagation problems in random media. This method simultaneously includes all features of spectral finite element and stochastic finite element methods, which leads to excellent accuracy and convergence by implementing Gauss–Lobatto–Legendre collocation points permitting to generate coarser meshes. In addition, the proposed StSFEM leads to diagonal mass matrices, which accelerates temporal integration schemes and provides desirable accuracy. Furthermore, it numerically solves the Fredholm integral equation arising from Karhunen–Loève Expansion with favorable accuracy and computing time. Here, the StSFEM is examined and developed to stochastic wave propagation phenomena through several numerical simulations. Results demonstrate successful performance of the StSFEM in the solved problems so that one can accomplish uncertainty quantification of time domain wave propagation within random continua by incorporating the StSFEM.

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.

A GPU domain decomposition solution for spectral stochastic finite element method

In intrusive methods for the stochastic analysis of engineering systems, the solution of the stochastic partial differential equations leads to an augmented algebraic system of equations, with respect to the corresponding deterministic problem. The solution of this augmented system can become quite challenging or even impossible due to the increased computational resources and effort required, especially in large-scale problems with large parameter variability. Recent developments in computer hardware regarding graphics processing units (GPU)-accelerated computing have been proven very promising due to their advanced capabilities and have been used in a number of scientific fields in order to enhance the solution of computationally high demanding problems. In this work, the benefits achieved with the exploitation of the GPU capabilities are demonstrated, in addressing intrusive stochastic mechanics problems where the solution of the resulting finite element algebraic equations is performed with the dual domain decomposition method, implementing specifically tailored preconditioners. The results show a significant enhancement of the spectral stochastic finite element method in terms of computational performance as well as of the consumed energy efficiency when applying the proposed methodology.