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.

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.

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.

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）.

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.

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.

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).

Deterministic and probabilistic analysis of great-depth braced excavations:A 32 m excavation case study in Paris

The Fort d’Issy-Vanves-Clamart(FIVC)braced excavation in France is analyzed to provide insights into the geotechnical serviceability assessment of excavations at great depth within deterministic and probabilistic frameworks.The FIVC excavation is excavated at 32 m below the ground surface in Parisian sedimentary basin and a plane-strain finite element analysis is implemented to examine the wall deflections and ground surface settlements.A stochastic finite element method based on the polynomial chaos Kriging metamodel(MSFEM)is then proposed for the probabilistic analyses.Comparisons with field measurements and former studies are carried out.Several academic cases are then conducted to investigate the great-depth excavation stability regarding the maximum horizontal wall deflection and maximum ground surface settlement.The results indicate that the proposed MSFEM is effective for probabilistic analyses and can provide useful insights for the excavation design and construction.A sensitivity analysis for seven considered random parameters is then implemented.The soil friction angle at the excavation bottom layer is the most significant one for design.The soil-wall interaction effects on the excavation stability are also given.

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.

Fuzzy stochastic damage mechanics(FSDM) based on fuzzy auto-adaptive control theory

In order to fully interpret and describe damage mechanics, the origin and development of fuzzy stochastic damage mechanics were introduced based on the analysis of the harmony of damage, probability, and fuzzy membership in the interval of [0,1]. In a complete normed linear space, it was proven that a generalized damage field can be simulated through β probability distribution. Three kinds of fuzzy behaviors of damage variables were formulated and explained through analysis of the generalized uncertainty of damage variables and the establishment of a fuzzy functional expression. Corresponding fuzzy mapping distributions, namely, the half-depressed distribution, swing distribution, and combined swing distribution, which can simulate varying fuzzy evolution in diverse stochastic damage situations, were set up. Furthermore, through demonstration of the generalized probabilistic characteristics of damage variables, the cumulative distribution function and probability density function of fuzzy stochastic damage variables, which show β probability distribution, were modified according to the expansion principle. The three-dimensional fuzzy stochastic damage mechanical behaviors of the Longtan rolled-concrete dam were examined with the self-developed fuzzy stochastic damage finite element program. The statistical correlation and non-normality of random field parameters were considered comprehensively in the fuzzy stochastic damage model described in this paper. The results show that an initial damage field based on the comprehensive statistical evaluation helps to avoid many difficulties in the establishment of experiments and numerical algorithms for damage mechanics analysis.

Fiber Concrete under Temperature Drop Load with Stochastic FEM

Plain concrete plate and fiber concrete plate subjected to temperature drop load were analyzed on stochastic finite element method (FEM).It is found that fibers can enhance concrete ability to resist tem- perature drop load for improving concrete's fracture energy and deferring the crack process.It is found for concrete not to improve apparently its tensile strength and fracture energy is recommended to be its appraisal parameter.

Probabilistic durability assessment approach of deteriorating RC bridges

A stochastic finite element computational methodology for probabilistic durability assessment of deteriorating reinforced concrete（RC） bridges by considering the time-and space-dependent variabilities is presented.First,finite element analysis with a smeared cracking approach is implemented.The time-dependent bond-slip relationship between steel and concrete,and the stress-strain relationship of corroded steel bars are considered.Secondly,a stochastic finite element-based computational framework for reliability assessment of deteriorating RC bridges is proposed.The spatial and temporal variability of several parameters affecting the reliability of RC bridges is considered.Based on the data reported by several researchers and from field investigations,the Monte Carlo simulation is used to account for the uncertainties in various parameters,including local and general corrosion in rebars,concrete cover depth,surface chloride concentration,chloride diffusion coefficient,and corrosion rate.Finally,the proposed probabilistic durability assessment approach and framework are applied to evaluate the time-dependent reliability of a girder of a RC bridge located on the Tianjin Binhai New Area in China.

Influences of Randomly Distributed Wall Thickness of Beverage Can on Its Strength and Stiffness

This paper describes the research undertaken on the strength and stiffness of fluctuation on the wall thickness of steel beverage cans using the Monte Carlo stochastic finite element method. Sample distributions were firstly assumed and then proven using the data observations of the wall thickness, the APDL language was then applied, and the stresses and displacements of the can were calculated by using the ANSYS software. It is concluded that the structural reliability of a steel making beverage can be estimated accurately.

Approach to minish scattering of results for split Hopkinson pressure bar test

Split Hopkinson pressure bar（SHPB） apparatus, usually used for testing behavior of material in median and high strain-rate, is now widely used in the study of rock dynamic constitutive relation, damage evolvement mechanism and energy consumption. However, the possible reasons of sampling disturbance, machining error and so on often lead to the scattering of test results, and bring ultimate difficulty for forming general test conclusion. Based on the stochastic finite element method, the uncertain parameters of specimen density ps, specimen radius Rs, specimen elastic modulus Es and specimen length Ls in the data processing of SHPB test were considered, and the correlation between the parameters and the test results was analyzed. The results show that the specimen radius Rs has direct correlation with the test result, improving the accuracy in preparing and measuring of specimen is an effective way to improve the accuracy of test and minish the scattering of results for SHPB test.

Effect of strength matching on the reliability of welded pipe with circumferential surface crack

For different strength matching, the reliability index and failure probability of welded pressure pipe with circumferential surface crack were calculated using three dimensional stochastic finite element method. This method has overcome the shortcomings of conservative results in safety assessment with deterministic fracture mechanics method. The effects of external moment and the depth of the circumferential surface crack (a) on the reliability of pressure pipe were also calculated and discussed. The calculation results indicate that the strength matching has certain effect on the reliability of the welded pressure pipe with circumferential surface crack. The failure probability of welded pressure pipe with high strength matching is lower than that with low strength matching at the same conditions. The effects of strength matching on the failure probability and reliability index increased by adding external moment (M) and the depth of the circumferential surface crack (a).

Quasi-Monte Carlo Simulation-Based SFEM for Slope Reliability Analysis

Considering the stochastic spatial variation of geotechnical parameters over the slope, a Stochastic Finite Element Method (SFEM) is established based on the combination of the Shear Strength Reduction (SSR) concept and quasi-Monte Carlo simulation. The shear strength reduction FEM is superior to the slice method based on the limit equilibrium theory in many ways, so it will be more powerful to assess the reliability of global slope stability when combined with probability theory. To illustrate the performance of the proposed method, it is applied to an example of simple slope. The results of simulation show that the proposed method is effective to perform the reliability analysis of global slope stability without presupposing a potential slip surface.

A Discussion on Two Stochastic Elliptic Modeling Strategies

Based on the study of two commonly used stochastic elliptic models:I:−∇·(a(x,w)·∇u(x,w))=f(x)and II:−∇·(a(x,w)⋄∇u(x,w))=f(x),we constructed a new stochastic elliptic model III:−∇·(a^(−1))^(⋄(−1))⋄∇u(x,w)
=f(x),in[20].The difference between models I and II is twofold:a scaling factor induced by the way of applying theWick product and the regularization induced by theWick product itself.In[20],we showed that model III has the same scaling factor as model I.In this paper we present a detailed discussion about the difference between models I and III with respect to the two characteristic parameters of the random coefficient,i.e.,the standard deviation s and the correlation length lc.Numerical results are presented for both one-and twodimensional cases.

Static Force Reliability Analysis of Truss Structure with Piezoelectric Patches Affixed to Its Surface

For the truss structure composed of active-elements with piezoelectric patches affixed to its surface,taking the mechani-cal-electric coupling effect under the action of electric loads and mechanical loads into consideration,the finite element model for static force analysis is established by using the theory of mechanics. The failure mechanism of piezoelectric elements is discussed and the failure criteria of piezoelectric elements are proposed. The expression of safety margins for the element of piezoelec...

Effects of Crack Opening Angle and External Loads on Reliability of Welded Pipe with Circumferential Crack

The reliability of welded pressure pipe with circumferential surface crack was calculated by using 3D stochastic finite element method.This method has overcome the shortcomings of conservative results in safety assessment with deterministic fracture mechanics method.The calculation of reliability was based on 3D elasticplastic stochastic finite element program which was developed by ourselves.The effects of variables such as fracture toughness,bending moment and depth of the circumferential surface crack on the structure reliability were also discussed.The calculation results indicate that the crack opening angle has certain effect on the reliability of the welded pipe.When the mean value of bending moment is 10kN·m,with the crack opening angle increasing from 30°to 180°,the failure probability of the welded pipe changes from 3.8379×10-8to 3.6943×10-6.When the mean value of bending moment is 15kN·m,with the crack opening angle increasing from 30°to 180°,the failure probability of the welded pipe changes from 3.986 5×10-6to 1.936 7×10-3.The bending moment has great effect on the reliability of the welded pipe.When the mean value of moment is changed from 10 to 15 kN·m,the failure probability of the welded pipe increases dramatically for the same circumferential crack opening angle.Regardless of changing of moment,the pipe has higher reliability if the crack opening angle is less than 60°.The method has put forward a new way for safety assessment of welded pipe with circumferential surface crack.

Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients.These problems model the conversion of starch into sugars in growing apples.The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization.This method leads to high-order accurate stochastic solutions.A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method.After Newton linearization,a point-based algebraic multigrid solution method is applied.In order to decrease the computational cost,alternative multigrid preconditioners are presented.Numerical results demonstrate the convergence properties,robustness and efficiency of the proposed multigrid methods.