Neumann stochastic finite element method for calculating temperature field of frozen soil based on random field theory

To study the effect of uncertain factors on the temperature field of frozen soil, we propose a method to calculate the spatial average variance from just the point variance based on the local average theory of random fields. We model the heat transfer coefficient and specific heat capacity as spatially random fields instead of traditional random variables. An analysis for calculating the random temperature field of seasonal frozen soil is suggested by the Neumann stochastic finite element method, and here we provide the computational formulae of mathematical expectation, variance and variable coefficient. As shown in the calculation flow chart, the stochastic finite element calculation program for solving the random temperature field, as compiled by Matrix Laboratory （MATLAB） sottware, can directly output the statistical results of the temperature field of frozen soil. An example is presented to demonstrate the random effects from random field parameters, and the feasibility of the proposed approach is proven by compar- ing these results with the results derived when the random parameters are only modeled as random variables. The results show that the Neumann stochastic finite element method can efficiently solve the problem of random temperature fields of frozen soil based on random field theory, and it can reduce the variability of calculation results when the random parameters are modeled as spatial- ly random fields.

Interval Analysis of the Finite Element Method for Stochastic Structures

A random parameter can be transformed into an interval number in the structural analysis with the concept of the confidence interval. Hence, analyses of uncertain structural systems can be used in the traditional FEM software. In some cases, the amount of solutions in stochastic structures is nearly as many as that in the traditional structural problems. In addition, a new method to evaluate the failure probability of structures is presented for the needs of the modern engineering design.

Stochastic Finite Element Method for Mechanical Vibration Based on Conjugate Gradient(CG)

When material properties, geometry parameters and applied loads are assumed to be stochastic, the vibration equation of a system is transformed to static problem by using Newmark method. In order to improve the computational efficiency and to save storage, the Conjugate Gradient （CG） method is presented. The CG is an effective method for solving a large system of linear equations and belongs to the method of iteration with rapid convergence and high precision. An example is given and calculated results are compared to validate the proposed methods.

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

Stochastic Finite Element Analysis of Plate and Shell Construction

The response of random plate and shell construction is analyzed with the stochastic finite element method (SFEM). Random material properties and geometric dimensions of construction are involved in this paper. A simplified isoparametric local average model is used to describe the random field. Numerical results of the examples indicate that the approach presented herein is an economical and efficient solution for such an analysis compared with Monte Carlo simulation (MCS).

A METHOD OF SOLVING THE FUZZY FINITE ELEMENT EQUATIONS IN MONOSOURCE FUZZY NUMBERS

For same cases the rules of monosource fuzzy numbers con be used into the solution of fuzzy stochastic finite element equations in engineering. This method can reduce the computing quantity of the solution. It can be proved that the amount of the solution is nearly as much as that with the general stochastic finite element method (SFEM). In addition, a new method to appreciate the structural fuzzy failure probability is presented for the needs of the modem engineering design.

Implementing the Node Based Smoothed Finite Element Method as User Element in Abaqus for Linear and Nonlinear Elasticity

In this paper,the node based smoothed-strain Abaqus user element(UEL)in the framework of finite element method is introduced.The basic idea behind of the node based smoothed finite element(NSFEM)is that finite element cells are divided into subcells and subcells construct the smoothing domain associated with each node of a finite element cell[Liu,Dai and Nguyen-Thoi(2007)].Therefore,the numerical integration is globally performed over smoothing domains.It is demonstrated that the proposed UEL retains all the advantages of the NSFEM,i.e.,upper bound solution,overly soft stiffness and free from locking in compressible and nearly-incompressible media.In this work,the constant strain triangular(CST)elements are used to construct node based smoothing domains,since any complex two dimensional domains can be discretized using CST elements.This additional challenge is successfully addressed in this paper.The efficacy and robustness of the proposed work is obtained by several benchmark problems in both linear and nonlinear elasticity.The developed UEL and the associated files can be downloaded from https://github.com/nsundar/NSFEM.

RELIABILITY OF ELASTO-PLASTIC STRUCTURE USING FINITE ELEMENT METHOD

A solution of probabilistic FEM for elastic-plastic materials is presented based on the incremental theory of plasticity and a modified initial stress method. The formulations are deduced through a direct differentiation scheme. Partial differentiation of displacement, stress and the performance function can be iteratively performed with the computation of the mean values of displacement and stress. The presented method enjoys the efficiency of both the perturbation method and the finite difference method, but avoids the approximation during the partial differentiation calculation. In order to improve the efficiency, the adjoint vector method is introduced to calculate the differentiation of stress and displacement with respect to random variables. In addition, a time-saving computational method for reliability index of elastic-plastic materials is suggested based upon the advanced First Order Second Moment (FOSM) and by the usage of Taylor expansion for displacement. The suggested method is also applicable to 3-D cases.

Meshfree Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

A four-node quadrilateral shell element with smoothed membrane-bending based on Mindlin-Reissner theory is proposed. The element is a combination of a plate bending and membrane element. It is based on mixed interpolation where the bending and membrane stiffness matrices are calculated on the boundaries of the smoothing cells while the shear terms are approximated by independent interpolation functions in natural coordinates. The proposed element is robust, computationally inexpensive and free of locking. Since the integration is done on the element boundaries for the bending and membrane terms, the element is more accurate than the MITC4 element for distorted meshes. This will be demonstrated for several numerical examples.

A Theory and Method for Modeling of Structures with Stochastic Parameters

In order to reflect the stochastic characteristics of structures more comprehensively and accurately, a theory and method for modeling of structures with stochastic parameters is presented by using probability finite element method and stochastic experiment data of structures based on the modeling of structures with deterministic parameters. Double-decker space frame is taken as an example to validate this theory and method, good results are gained.

基于Stochastic Kriging的柔性机翼稳健性优化设计

采用随机代理模型方法对柔性机翼气动外形进行稳健性优化设计。相比确定性优化设计,稳健性设计能够考虑设计变量和参数的扰动,保持设计结果在不确定性影响下的性能稳定。采用高精度的气动/结构耦合求解器(耦合Navier-Stokes方程和结构静力学方程)分析柔性机翼的变形情况和气动效率。为了提高优化效率,建立随机Kriging(Stochastic Kriging,SK)代理模型,将确定性的Kriging代理模型发展到随机空间,通过有限次输入得到数据的固有不确定性。对柔性M6机翼的气动外形进行稳健性优化设计,结果表明:相比确定性代理模型的稳健性优化结果,应用随机代理模型的优化结果的设计点阻力系数减小2.8 counts,在可变马赫数范围内阻力系数均值减小3.2 counts,优化结果具有较高的设计点气动效率和阻力发散特性,并且优化后构型的翼根弯矩有明显减小,体现随机代理模型在稳健性优化设计系统中的优势,同时也说明建立的SK代理模型具有较高的预测精度。

FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD FOR STOCHASTIC HELMHOLTZ EQUATION IN TWO-AND THREE-DIMENSIONS

In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d （d = 2, 3）. Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.

ERROR ESTIMATES OF FINITE ELEMENT METHODS FOR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.

Fuzzy stochastic generalized reliability studies on embankment systems based on first-order approximation theorem

In order to address the complex uncertainties caused by interfacing between the fuzziness and randomness of the safety problem for embankment engineering projects, and to evaluate the safety of embankment engineering projects more scientifically and reasonably, this study presents the fuzzy logic modeling of the stochastic finite element method （SFEM） based on the harmonious finite element （HFE） technique using a first-order approximation theorem. Fuzzy mathematical models of safety repertories were introduced into the SFEM to analyze the stability of embankments and foundations in order to describe the fuzzy failure procedure for the random safety performance function. The fuzzy models were developed with membership functions with half depressed gamma distribution, half depressed normal distribution, and half depressed echelon distribution. The fuzzy stochastic mathematical algorithm was used to comprehensively study the local failure mechanism of the main embankment section near Jingnan in the Yangtze River in terms of numerical analysis for the probability integration of reliability on the random field affected by three fuzzy factors. The result shows that the middle region of the embankment is the principal zone of concentrated failure due to local fractures. There is also some local shear failure on the embankment crust. This study provides a referential method for solving complex multi-uncertainty problems in engineering safety analysis.

Stochastic Response Analysis of Piled Offshore Platforms to Earthquake Load

In this paper, using the theory of stochastic analysis of the response to earthquake load, a stochastic analysis method of the response of piled platforms to earthquake load has been established. In the method, the strong ground motion is considered as three dimensional stationary white noise process and the pile-soil interaction and water-structure interaction are considered. The stochastic response of a typical platform to earthquake load has been computed with this method and the results compared with those obtained with the response spectrum analysis method. The comparison shows that the stochastic analysis method of the response of piled platforms to earthquake load is suitable for this kind of analysis.

HIERARCHICAL STOCHASTIC FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSIS

In this paper, the hierarchical approach is adopted for series representation of the stochastic nodal displacement vector using the hierarchical basis vectors, while the Karhunen- Loire series expansion technique is employed to discretize the random field into a set of random variables. A set of hierarchical basis vectors are defined to approximate the stochastic response quantities. The stochastic variational principle instead of the projection scheme is adopted to develop a hierarchical stochastic finite element method （HSFEM） for stochastic structures under stochastic loads. Simplified expressions of coefficients of governing equations and the first two statistical moments of the response quantities in the schemes of the HSFEM are developed, so that the time consumed for computation can be greatly reduced. Investigation in this paper suggests that the HSFEM yields a series of stiffness equations with similar dimensionality as the perturbation stochastic finite element method （PSFEM）. Two examples are presented for numerical study on the performance of the HSFEM in elastic structural problems with stochastic Young＇s Modulus and external loads. Results show that the proposed method can achieve higher accuracy than the PSFEM for cases with large coefficients of variation, and yield results agreeing well with those obtained by the Monte Carlo simulation （MCS）.

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

This paper uses the spectral stochastic finite element method(SSFEM)for analyzing reinforced concrete(RC)beam/slab problems.In doing so,it presents a new framework to study how the correlation length of a random field(RF)with uncertain parameters will affect modeling uncertainties and reliability evaluations.It considers:1)different correlation lengths for uncertainty parameters,and 2)dead and live loads as well as the elasticity moduli of concrete and steel as a multi-dimensional RF in concrete structures.To show the SSFEM’s efficiency in the study of concrete structures and to evaluate the sensitivity of the correlation length effects in evaluating the reliability,two examples of RC beams and slabs have been investigated.According to the results,the RF correlation length is effective in modeling uncertainties and evaluating reliabilities;the longer the correlation length,the greater the dispersion range of the structure response and the higher the failure probability.

Finite Element Methods for Nonlinear Backward Stochastic Partial Differential Equations and Their Error Estimates

In this paper,we consider numerical approximation of a class of nonlinear backward stochastic partial differential equations(BSPDEs).By using finite element methods in the physical space domain and the Euler method in the time domain,we propose a spatial finite element semi-discrete scheme and a spatio-temporal full discrete scheme for solving the BSPDEs.Errors of the schemes are rigorously analyzed and theoretical error estimates with convergence rates are obtained.

Three-dimensional stochastic seepage field for embankment engineering

Owing to the complexity of geo-engineering seepage problems influenced by different random factors, three-dimensional simulation and analysis of the stochastic seepage field plays an important role in engineering applications. A three-dimensional anisotropic heterogeneous steady random seepage model was developed on the basis of the finite element method. A statistical analysis of the distribution characteristics of soil parameters sampled from the main embankment of the Yangtze River in the Southern Jingzhou zone of China was conducted. The Kolomogorov-Smirnov test verified the statistical hypothesis that the permeability coefficient tensor has a Gaussian distribution. With the help of numerical analysis of the stochastic seepage field using the developed model, various statistical and random characteristics of the stochastic seepage field of the main embankment of the Yangtze River in the Southern Jingzhou zone of China were investigated. The model was also examined with statistical testing. Through the introduction of random variation of the upstream and downstream water levels into the model, the effects of the boundary randomness due to variation of the downstream and upstream water levels on the variation of simulated results presented with a vector series of the random seepage field were analyzed. Furthermore, the combined influence of the variation of the soil permeability coefficient and such seepage resistance measures as the cut-off wall and relief ditch on the hydraulic head distribution was analyzed and compared with the results obtained by determinate analysis. Meanwhile, sensitivities of the hydraulic gradient and downstream exit height to the variation of boundary water level were studied. The validity of the simulated results was verified by stochastic testing and measured data. The developed model provides more detail and a full stochastic algorithm to characterize and analyze three-dimensional stochastic seepage field problems.