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Journal articles on the topic 'Spectral stochastic finite element'

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Author: Grafiati

Published: 6 September 2023

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1

Ghanem, Roger G., and Pol D. Spanos. "Spectral Stochastic Finite‐Element Formulation for Reliability Analysis." Journal of Engineering Mechanics 117, no. 10 (October 1991): 2351–72. http://dx.doi.org/10.1061/(asce)0733-9399(1991)117:10(2351).

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2

Ghanem, Roger G., and Robert M. Kruger. "Numerical solution of spectral stochastic finite element systems." Computer Methods in Applied Mechanics and Engineering 129, no. 3 (January 1996): 289–303. http://dx.doi.org/10.1016/0045-7825(95)00909-4.

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3

Honda, Riki, Ghanem Roger, and Michihiro KITAHARA. "Spectral Stochastic Finite Element Method for Log-Normal Uncertainty." Journal of applied mechanics 7 (2004): 391–98. http://dx.doi.org/10.2208/journalam.7.391.

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4

Sousedík, Bedřich, and Howard C. Elman. "Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods." SIAM/ASA Journal on Uncertainty Quantification 4, no. 1 (January 2016): 163–89. http://dx.doi.org/10.1137/140999359.

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5

Gaignaire, R., S. Clnet, B. Sudret, and O. Moreau. "3-D Spectral Stochastic Finite Element Method in Electromagnetism." IEEE Transactions on Magnetics 43, no. 4 (April 2007): 1209–12. http://dx.doi.org/10.1109/tmag.2007.892300.

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6

Powell, C. E., and H. C. Elman. "Block-diagonal preconditioning for spectral stochastic finite-element systems." IMA Journal of Numerical Analysis 29, no. 2 (April 2, 2008): 350–75. http://dx.doi.org/10.1093/imanum/drn014.

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7

Chen, Nian-Zhong, and C. Guedes Soares. "Spectral stochastic finite element analysis for laminated composite plates." Computer Methods in Applied Mechanics and Engineering 197, no. 51-52 (October 2008): 4830–39. http://dx.doi.org/10.1016/j.cma.2008.07.003.

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8

Chung, Doo Bo, Miguel A. Gutiérrez, Lori L. Graham-Brady, and Frederik-Jan Lingen. "Efficient numerical strategies for spectral stochastic finite element models." International Journal for Numerical Methods in Engineering 64, no. 10 (2005): 1334–49. http://dx.doi.org/10.1002/nme.1404.

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9

Beddek, K., S. Clénet, O. Moreau, and Y. Le Menach. "Spectral stochastic finite element method for solving 3D stochastic eddy current problems." International Journal of Applied Electromagnetics and Mechanics 39, no. 1-4 (September 5, 2012): 753–60. http://dx.doi.org/10.3233/jae-2012-1539.

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10

Ghanem, R. G., and P. D. Spanos. "Spectral techniques for stochastic finite elements." Archives of Computational Methods in Engineering 4, no. 1 (March 1997): 63–100. http://dx.doi.org/10.1007/bf02818931.

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11

Adhikari, Sondipon. "Doubly Spectral Stochastic Finite-Element Method for Linear Structural Dynamics." Journal of Aerospace Engineering 24, no. 3 (July 2011): 264–76. http://dx.doi.org/10.1061/(asce)as.1943-5525.0000070.

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12

Ghanem, R., and M. Pellissetti. "Adaptive data refinement in the spectral stochastic finite element method." Communications in Numerical Methods in Engineering 18, no. 2 (January 10, 2002): 141–51. http://dx.doi.org/10.1002/cnm.476.

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13

Sheu, Guang Yih. "High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations." Applied Mathematics 04, no. 05 (2013): 18–28. http://dx.doi.org/10.4236/am.2013.45a003.

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14

Ghanem, R. "Higher-Order Sensitivity of Heat Conduction Problems to Random Data Using the Spectral Stochastic Finite Element Method." Journal of Heat Transfer 121, no. 2 (May 1, 1999): 290–99. http://dx.doi.org/10.1115/1.2825979.

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The spectral formulation of the stochastic finite element method is applied to the problem of heat conduction in a random medium. Specifically, the conductivity of the medium, as well as its heat capacity are treated as uncorrelated random processes with spatial random fluctuations. This paper introduces the basic concepts of the spectral stochastic finite element method using a simple one-dimensional heat conduction examples. The implementation of the method is demonstrated for both Gaussian and log-normal material properties. Moreover, the case of the material properties being modeled as random variables is presented as a simple digression of the formulation for the stochastic process case. Both Gaussian and log-normal models for the material properties are treated.

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15

Hang, D. T., X. T. Nguyen, and D. N. Tien. "Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements with Discretization Random Field by the Point Method." Engineering, Technology & Applied Science Research 12, no. 2 (April 9, 2022): 8458–62. http://dx.doi.org/10.48084/etasr.4819.

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This study examined the discretization random field of the elastic modulus by a point method to develop a stochastic finite element method for the stochastic buckling of a non-uniform column. The formulation of stochastic analysis of a non-uniform column was constructed using the perturbation method in conjunction with the finite element method. The spectral representation was used to generate a random field to employ the Monte Carlo simulation for validation with a stochastic finite element approach. The results of the stochastic buckling problem of non-uniform columns with the random field of elastic modulus by comparing the first-order perturbation technique were in good agreement with those obtained from the Monte Carlo simulation. The numerical results showed that the response of the coefficient of variation of critical loads increased when the ratio of the correlation distance of the random field increased.

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16

Lehikoinen, Antti. "Spectral Stochastic Finite Element Method for Electromagnetic Problems with Random Geometry." Electrical, Control and Communication Engineering 6, no. 1 (October 23, 2014): 5–12. http://dx.doi.org/10.2478/ecce-2014-0011.

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Abstract In electromagnetic problems, the problem geometry may not always be exactly known. One example of such a case is a rotating machine with random-wound windings. While spectral stochastic finite element methods have been used to solve statistical electromagnetic problems such as this, their use has been mainly limited to problems with uncertainties in material parameters only. This paper presents a simple method to solve both static and time-harmonic magnetic field problems with source currents in random positions. By using an indicator function, the geometric uncertainties are effectively reduced to material uncertainties, and the problem can be solved using the established spectral stochastic procedures. The proposed method is used to solve a demonstrative single-conductor problem, and the results are compared to the Monte Carlo method. Based on these simulations, the method appears to yield accurate mean values and variances both for the vector potential and current, converging close to the results obtained by time-consuming Monte Carlo analysis. However, further study may be needed to use the method for more complicated multi-conductor problems and to reduce the sensitivity of the method on the mesh used.

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17

Gaignaire, R., S. Clenet, O. Moreau, and B. Sudret. "Current Calculation in Electrokinetics Using a Spectral Stochastic Finite Element Method." IEEE Transactions on Magnetics 44, no. 6 (June 2008): 754–57. http://dx.doi.org/10.1109/tmag.2008.915801.

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18

Stavroulakis, G., D. G. Giovanis, V. Papadopoulos, and M. Papadrakakis. "A GPU domain decomposition solution for spectral stochastic finite element method." Computer Methods in Applied Mechanics and Engineering 327 (December 2017): 392–410. http://dx.doi.org/10.1016/j.cma.2017.08.042.

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19

Jin, Bangti, and Jun Zou. "Inversion of Robin coefficient by a spectral stochastic finite element approach." Journal of Computational Physics 227, no. 6 (March 2008): 3282–306. http://dx.doi.org/10.1016/j.jcp.2007.11.042.

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20

Adhikari, S. "A reduced spectral function approach for the stochastic finite element analysis." Computer Methods in Applied Mechanics and Engineering 200, no. 21-22 (May 2011): 1804–21. http://dx.doi.org/10.1016/j.cma.2011.01.015.

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21

Chakraborty, Subrata, and Santi Sekhar Dey. "Stochastic Finite Element Simulation of Uncertain Structures Subjected to Earthquake." Shock and Vibration 7, no. 5 (2000): 309–20. http://dx.doi.org/10.1155/2000/730364.

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In present study, the stochastic finite element simulation based on the efficient Neumann expansion technique is extended for the analysis of uncertain structures under seismically induced random ground motion. The basic objective is to investigate the possibility of applying the Neumann expansion technique coupled with the Monte Carlo simulation for dynamic stochastic systems upto that extent of parameter variation after which the method is no longer gives accurate results compared to that of the direct Monte carlo simulation. The stochastic structural parameters are discretized by the local averaging method and then simulated by Cholesky decomposition of the respective covariance matrix. The earthquake induced ground motion is treated as stationary random process defined by respective power spectral density function. Finally, the finite element solution has been obtained in frequency domain utilizing the advantage of Neumann expansion technique.

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22

Nguyen-Van, Thuan, and Thanh Bui-Tien. "Investigation of the Eigenvector of Stochastic Finite Element Methods of Functionally Graded Beams with Random Elastic Modulus." Engineering, Technology & Applied Science Research 13, no. 4 (August 9, 2023): 11253–57. http://dx.doi.org/10.48084/etasr.5991.

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This paper presents a stochastic finite element method to calculate the variation of eigenvalues and eigenvectors of functionally graded beams. The modulus of functionally graded material is assumed to have spatial uncertainty as a one-dimensional random field. The formulation of the stochastic finite element method for the functionally graded beam due to the randomness of the elastic modulus of the beam is given using the first-order perturbation approach. This approach was validated with Monte Carlo simulation in previous studies using spectral representation to generate the random field. The statistics of the beam responses were investigated using the first-order perturbation method for different fluctuations of the elastic modulus. A comparison of the results of the stochastic finite element method with the first-order perturbation approach and the Monte Carlo simulation showed a minimal difference.

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23

Hussein, A., M. El-Tawil, W. El-Tahan, and A. A. Mahmoud. "Solution of randomly excited stochastic differential equations with stochastic operator using spectral stochastic finite element method (SSFEM)." Structural Engineering and Mechanics 28, no. 2 (January 30, 2008): 129–52. http://dx.doi.org/10.12989/sem.2008.28.2.129.

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24

Ghosh, Debraj. "Probabilistic Interpretation of Conjugate Gradient Iterations in Spectral Stochastic Finite Element Method." AIAA Journal 52, no. 6 (June 2014): 1313–16. http://dx.doi.org/10.2514/1.j052769.

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25

Sepahvand, K., and S. Marburg. "Spectral stochastic finite element method in vibroacoustic analysis of fiber-reinforced composites." Procedia Engineering 199 (2017): 1134–39. http://dx.doi.org/10.1016/j.proeng.2017.09.241.

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26

Yazdani, Abbas, Hamed Ghohani Arab, and Mohsen Rashki. "Simplified spectral stochastic finite element formulations for uncertainty quantification of engineering structures." Structures 28 (December 2020): 1924–45. http://dx.doi.org/10.1016/j.istruc.2020.09.040.

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27

Giovanis, Dimitris G., Vissarion Papadopoulos, and George Stavroulakis. "An adaptive spectral Galerkin stochastic finite element method using variability response functions." International Journal for Numerical Methods in Engineering 104, no. 3 (April 27, 2015): 185–208. http://dx.doi.org/10.1002/nme.4926.

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28

Noori, M. S. M., and R. M. Abbas. "Reliability Analysis of an Uncertain Single Degree of Freedom System Under Random Excitation." Engineering, Technology & Applied Science Research 12, no. 5 (October 2, 2022): 9252–57. http://dx.doi.org/10.48084/etasr.5193.

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In practical engineering problems, uncertainty exists not only in external excitations but also in structural parameters. This study investigates the influence of structural geometry, elastic modulus, mass density, and section dimension uncertainty on the stochastic earthquake response of portal frames subjected to random ground motions. The North-South component of the El Centro earthquake in 1940 in California is selected as the ground excitation. Using the power spectral density function, the two-dimensional finite element model of the portal frame’s base motion is modified to account for random ground motions. A probabilistic study of the portal frame structure using stochastic finite elements utilizing Monte Carlo simulation is presented using the finite element program ABAQUS. The dynamic reliability and probability of failure of stochastic and deterministic structures based on the first-passage failure were examined and evaluated. The results revealed that the probability of failure increases due to the randomness of stiffness and mass of the structure. The influence of uncertain parameters on reliability analysis depends on the extent of variance in structural parameters.

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29

Ta Duy, Hien, Nguyen Dang Diem, Giap Van Tan, Vu Van Hiep, and Nguyen Van Thuan. "Stochastic Higher-order Finite Element Model for the Free Vibration of a Continuous Beam resting on Elastic Support with Uncertain Elastic Modulus." Engineering, Technology & Applied Science Research 13, no. 1 (February 5, 2023): 9985–90. http://dx.doi.org/10.48084/etasr.5456.

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This paper deals with a continuous beam resting on elastic support with elastic modulus derived from a random process. Governing equations of the stochastic higher-order finite element method of the free vibration of the continuous beam were derived from Hamilton's principle. The random process of elastic modulus was discretized by averaging random variables in each element. A solution for the stochastic eigenvalue problem for the free vibration of the continuous beam was obtained by using the perturbation technique, in conjunction with the finite element method. Spectral representation was used to generate a random process and employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.

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30

NAKAGAWA, Hidenori, and Muneo HORI. "APPLICATION OF NONLINEAR SPECTRAL STOCHASTIC FINITE ELEMENT METHOD TO SURFACE EARTHQUAKE FAULT PROBLEMS." Journal of Japan Society of Civil Engineers, Ser. A1 (Structural Engineering & Earthquake Engineering (SE/EE)) 67, no. 2 (2011): 225–41. http://dx.doi.org/10.2208/jscejseee.67.225.

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31

HONDA, Riki. "Analysis of Wave Propagation in Random Media by Spectral Stochastic Finite Element Method." Doboku Gakkai Ronbunshu, no. 689 (2001): 321–31. http://dx.doi.org/10.2208/jscej.2001.689_321.

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32

Ning Liu and Guang Ting Liu. "Spectral stochastic finite element analysis of periodic random thermal creep stress in concrete." Engineering Structures 18, no. 9 (September 1996): 669–74. http://dx.doi.org/10.1016/0141-0296(96)00015-6.

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33

Zakian, P., and N. Khaji. "A stochastic spectral finite element method for wave propagation analyses with medium uncertainties." Applied Mathematical Modelling 63 (November 2018): 84–108. http://dx.doi.org/10.1016/j.apm.2018.06.027.

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34

Beavers, J., K. Huddleston, N. Hines, and W. McNeil. "Modeling electron transport and multiplication in photomultiplier tubes using COMSOL Multiphysics^®." Journal of Instrumentation 17, no. 12 (December 1, 2022): P12015. http://dx.doi.org/10.1088/1748-0221/17/12/p12015.

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Abstract Combining stochastic and finite element methods, a modeling approach was executed that will inform new photomultiplier tube and scintillation detector designs. Time-dependent signal formation within a commercially available photomultiplier tube was modeled including the release and transport of electrons from the photocathode through the dynode stages. An ET Enterprises 9214B photomultiplier tube was digitally reproduced using Computed Tomography, X-ray radiography, and SolidWorks solid-modeling software. Simulations were executed with COMSOL Multiphysics® finite element solving package. Stochastic models of electron emission from the photocathode and dynodes were integrated within the COMSOL framework. Photoelectron emission energy was modeled by combining NaI(Tl) spectral emission characteristics and K2CsSb photocathode quantum efficiency. Secondary electron emission yields were produced to follow nominal photomultiplier gain, while secondary electron energies were sampled from the Chung-Everhart distribution. Electron emission trajectories were sampled according to Lambert's cosine law. Coupling stochastic and finite element models, simulation reproduced signal formation for the commercial photomultiplier tube including timing characteristics within 9.5% and gain within 3% over a voltage range of 900–1250 V.

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35

Wu, Yuching, and Jianzhuang Xiao. "The Multiscale Spectral Stochastic Finite Element Method for Chloride Diffusion in Recycled Aggregate Concrete." International Journal of Computational Methods 15, no. 01 (September 27, 2017): 1750078. http://dx.doi.org/10.1142/s0219876217500785.

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In this study, the multiscale stochastic finite element method (MsSFEM) was developed based on a novel digital image kernel to make analysis for chloride diffusion in recycled aggregate concrete (RAC). It is significant to study the chloride diffusivity in RAC, because when RAC was applied in coastal areas, chloride-induced rebar corrosion became a common problem for concrete infrastructures. The MsSFEM was an efficient tool to examine the effect of microscopic randomness of RAC on the chloride diffusivity. Based on the proposed digital image kernel, the Karhunen–Loeve expansion and the polynomial chaos were used in the stochastic homogenization process. To investigate advantages and disadvantages of both generation and application of the proposed digital image kernel, it was compared with many other kernels. The comparisons were made between the method to develop the digital image kernel, which is called the pixel-matrix method, and other methods, and between the application of the kernel and various other kernels. It was shown that the proposed digital image kernel is superior to other kernels in many aspects.

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36

Appalanaidu, Y., Anindya Roy, and Sayan Gupta. "Stochastic Creep Damage Estimation in Pipings with Spatial Non-gaussian Uncertainties Using Spectral Stochastic Finite Element Method." Procedia Engineering 86 (2014): 677–84. http://dx.doi.org/10.1016/j.proeng.2014.11.069.

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37

Noori, Mohammed S. M., and Rafaa M. Abbas. "Dynamic Response and Reliability Analysis of Stochastic Multi-Story Frame Structures under Random Excitation." Mathematical Modelling of Engineering Problems 9, no. 5 (December 13, 2022): 1335–42. http://dx.doi.org/10.18280/mmep.090523.

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In earthquake engineering problems, uncertainty exists not only in the seismic excitations but also in the structure's parameters. This study investigates the influence of structural geometry, elastic modulus, mass density, and section dimension uncertainty on the stochastic earthquake response of a multi-story moment resisting frame subjected to random ground motion. The North-south component of the Ali Gharbi earthquake in 2012, Iraq, is selected as ground excitation. Using the power spectral density function (PSD), the two-dimensional finite element model of the moment resisting frame's base motion is modified to account for random ground motion. The probabilistic study of the moment resisting frame structure using stochastic finite element utilizing Monte Carlo simulation was presented using the finite element program ABAQUS. The dynamic reliability and probability of failure of the stochastic and deterministic structure based on the first passage failure were checked and evaluated. Results revealed that the probability of failure increased due to randomness in stiffness and mass of the structure. Generally, natural frequencies for the lower modes of vibration and relative displacements for the lower stories were more sensitive to the randomness in system parameters.

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38

Khaji, N., and P. Zakian. "Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method." Mechanics of Advanced Materials and Structures 24, no. 12 (December 21, 2016): 1030–42. http://dx.doi.org/10.1080/15376494.2016.1202359.

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39

Beddek, Karim, Yvonnick Le Menach, Stephane Clenet, and Olivier Moreau. "3-D Stochastic Spectral Finite-Element Method in Static Electromagnetism Using Vector Potential Formulation." IEEE Transactions on Magnetics 47, no. 5 (May 2011): 1250–53. http://dx.doi.org/10.1109/tmag.2010.2076274.

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40

Sepahvand, K. "Spectral stochastic finite element vibration analysis of fiber-reinforced composites with random fiber orientation." Composite Structures 145 (June 2016): 119–28. http://dx.doi.org/10.1016/j.compstruct.2016.02.069.

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41

Ngah, M. F., and A. Young. "Application of the spectral stochastic finite element method for performance prediction of composite structures." Composite Structures 78, no. 3 (May 2007): 447–56. http://dx.doi.org/10.1016/j.compstruct.2005.11.009.

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42

Ghanem, R. "The Nonlinear Gaussian Spectrum of Log-Normal Stochastic Processes and Variables." Journal of Applied Mechanics 66, no. 4 (December 1, 1999): 964–73. http://dx.doi.org/10.1115/1.2791806.

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A procedure is presented in this paper for developing a representation of lognormal stochastic processes via the polynomial chaos expansion. These are processes obtained by applying the exponential operator to a gaussian process. The polynomial chaos expansion results in a representation of a stochastic process in terms of multidimensional polynomials orthogonal with respect to the gaussian measure with the dimension defined through a set of independent normalized gaussian random variables. Such a representation is useful in the context of the spectral stochastic finite element method, as well as for the analytical investigation of the mathematical properties of lognormal processes.

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43

Wu, Yuching, and Jianzhuang Xiao. "Digital-Image-Driven Stochastic Homogenization for Recycled Aggregate Concrete Based on Material Microstructure." International Journal of Computational Methods 16, no. 07 (July 26, 2019): 1850104. http://dx.doi.org/10.1142/s0219876218501049.

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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.

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44

Verhoosel, C. V., M. A. Gutiérrez, and S. J. Hulshoff. "Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems." International Journal for Numerical Methods in Engineering 68, no. 4 (2006): 401–24. http://dx.doi.org/10.1002/nme.1712.

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45

Ben Souf, Mohamed Amine, Mohamed Ichchou, Olivier Bareille, Noureddine Bouhaddi, and Mohamed Haddar. "Dynamics of random coupled structures through the wave finite element method." Engineering Computations 32, no. 7 (October 5, 2015): 2020–45. http://dx.doi.org/10.1108/ec-08-2014-0173.

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Purpose – The purpose of this paper is to develop a new formulation using spectral approach, which can predict the wave behavior to uncertain parameters in mid and high frequencies. Design/methodology/approach – The work presented is based on a hybridization of a spectral method called the “wave finite element (WFE)” method and a non-intrusive probabilistic approach called the “polynomial chaos expansion (PCE).” The WFE formulation for coupled structures is detailed in this paper. The direct connection with the conventional finite element method allows to identify the diffusion relation for a straight waveguide containing a mechanical or geometric discontinuity. Knowing that the uncertainties play a fundamental role in mid and high frequencies, the PCE is applied to identify uncertainty propagation in periodic structures with periodic uncertain parameters. The approach proposed allows the evaluation of the dispersion of kinematic and energetic parameters. Findings – The authors have found that even though this approach was originally designed to deal with uncertainty propagation in structures it can be competitive with its low time consumption. The Latin Hypercube Sampling (LHS) is also employed to minimize CPU time. Originality/value – The approach proposed is quite new and very simple to apply to any periodic structures containing variabilities in its mechanical parameters. The Stochastic Wave Finite Element can predict the dynamic behavior from wave sensitivity of any uncertain media. The approach presented is validated for two different cases: coupled waveguides with and without section modes. The presented results are verified vs Monte Carlo simulations.

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46

Farah, Khaled, Mounir Ltifi, and Hedi Hassis. "A Study of Probabilistic FEMs for a Slope Reliability Analysis Using the Stress Fields." Open Civil Engineering Journal 9, no. 1 (May 14, 2015): 196–206. http://dx.doi.org/10.2174/1874149501509010196.

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In this paper, the applicability and the effectiveness of the probabilistic finite element methods (FEMs) such as the perturbation method, and the Spectral Stochastic Finite Element Method (SSFEM) applied to the reliability analysis of the slope stability have been studied. The results were checked by the Monte Carlo simulation and a direct coupling ap-proach combining the deterministic finite elements code and First Order Reliability Method (FORM) algorithm. These methods are presented considering the spatial variation of soil strength parameters and Young modulus. The random field is used to describe the spatial variation. Also, the reliability analysis is conducted using a performance function formulat-ed in terms of the stochastic stress mobilized along the sliding surface. The present study shows that the perturbation method and SSFEM can be considered as practical methods to conduct a second moment analysis of the slope stability taking into account the spatial variability of soil properties since good results are obtained with acceptable estimated rela-tive errors. Finally, the perturbation method is performed to delimit the location of the critical probabilistic sliding surfac-es and to evaluate the effect of the correlation length of soil strength parameters on the safety factor. In addition, the two methods are used to estimate the probability density and the cumulative distribution function of the factor of safety.

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47

NAKAGAWA, Hidenori, and Muneo HORI. "Analysis of Ground Surface Deformation Caused by Strike-Slip Fault Using Spectral Stochastic Finite Element Method." Journal of applied mechanics 5 (2002): 573–80. http://dx.doi.org/10.2208/journalam.5.573.

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48

Zakian, P., and N. Khaji. "A novel stochastic-spectral finite element method for analysis of elastodynamic problems in the time domain." Meccanica 51, no. 4 (July 24, 2015): 893–920. http://dx.doi.org/10.1007/s11012-015-0242-9.

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49

Nouy, Anthony. "Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms." Computer Methods in Applied Mechanics and Engineering 197, no. 51-52 (October 2008): 4718–36. http://dx.doi.org/10.1016/j.cma.2008.06.012.

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50

Kundu, A., F. A. DiazDelaO, S. Adhikari, and M. I. Friswell. "A hybrid spectral and metamodeling approach for the stochastic finite element analysis of structural dynamic systems." Computer Methods in Applied Mechanics and Engineering 270 (March 2014): 201–19. http://dx.doi.org/10.1016/j.cma.2013.11.013.

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