Stochastic finite elements Research Papers (original) (raw)

Using the Stochastic Finite Element Method (SFEM) to perform reliability analysis of the free vibration of composite plates with material and fabrication uncertainties has received much attention lately. In this work the stochastic analysis is performed using the First-Order Reliability Method (FORM-method 2) and the Second-Order Reliability Method (SORM). The basic random variables include laminae stiffness properties and material density, as well as the randomness in ply orientation angles. Modeling of the composite behavior utilizes a nine-noded isoparametric Lagrangian element based on the third-order shear deformation theory. Calculating the eigenvectors at the mean values of the variables proves to be a reasonable simplification which significantly increases solution speed. The stochastic finite element code is validated using comparisons with results of Monte Carlo simulation technique with importance sampling. Results show that SORM is an excellent rapid tool in the stochastic analysis of free vibration of composite plates, when compared to the slower Monte Carlo simulation techniques.

- by
- •
- Mechanical Engineering, Civil Engineering, Computational Mechanics, Monte Carlo Simulation

Author of the monograph, Juraj Králik, has been working at the Department of
Structural Mechanics as assistant since September 1, 1976 and as associate professor since
January 18, 1988. During the years 2000 - 2006 he was the head of Department. He holds
lectures in two study programs: Engineering Structures and Transport Structures; and Civil
Engineering and Architecture. He teaches the following subjects: Mechanics of Structures and
Materials, Seismic Enginering and Computer Design, Risk Engineering, Safety and
Reliability of Buildings. His regular students and doctoral students have won several prizes in
the student research competitions at the Faculty. He has been implementing advanced
computer programs and methods in teaching and research at the Department, being himself an
author of more than 100 programs of the static and dynamic applications. He is guarantor of
using the licensed software systems ANSYS and MathCAD at the Department and Faculty,
too. As the co-guarantor, he has established the doctoral study program “Applied Mechanics”.
During the years 2000-2006, in cooperation with the Slovak Chamber of the Civil Engineers
he was the supervisor of seven volumes of the postgraduate study course „Aeroelasticity and
Seismicity“, and the chairman of seven volumes of the International Conference „New Trends
in the Statics and Dynamics of Buildings“.
His results were presented in more than 300 papers in conference proceedings and
journals, 10 papers are indexed in prestigious database „Web of Science“. Three papers were
published in the currented international journals „Mathematics and Computers in Simulation”
(1999), „Control and Cybernetics” (2006) and “Engineering Structures” (2009). 17 research
and grant projects were managed by him. His works were cited in more than 200 papers in
scientific and special publications. As the reputable scientific personality he was the member
of the scientific committees of several international conferences abroad. In the year 1989 he
was appointed an expert of the safety and reliability of nuclear power plants in Slovakia. He
cooperated at the analysis of the seismic resistance of the nuclear power plant buildings and
their safety under impact of explosion, missile and container drop. More than 100 expertises
were realized by him in the field of the safety and reliability of the NPP buildings in Slovakia.
Some of his research and expert works were awarded by significant institutions. The
most significant is the honorable award of the Czech Engineering Academy for the paper
„Probability Analysis of Reinforced Concrete Structure Failure of Nuclear Power Plants Due
to Loss of Coolant Accident “ published in the journal „ENGINEERING MECHANICS“ in
2006.

For accurate prediction of composite failure, microstructural variability must be
considered. The distribution of epoxy stiffness and hardness were determined by
nanoindentation and used in stochastic finite element modeling. Another key microstructural
feature, fiber volume fraction variability, was determined by image processing of an SEM
image. Stochastic failure analysis was implemented on a micromechanics model of hexagonal
fiber packing to predict the initiation of failure under multiaxial loadings. Three failure
criteria were employed for characterization of failure: maximum stress, von Mises, and
Christensen. Failure envelopes were developed for stochastic and average models. The results
revealed that the variability in epoxy strength influence the failure behavior significantly,
whereas, stiffness variability has minimal effect.

A 3-D explicit dynamic finite element analysis is performed
to determine the contact force and displacement between the impactor and the target. The uncertainties associated with the properties of the composite material, loading condition, and assessment of critical stresses affect the failure limit state to a greater extent. The Gaussian response surface method is used to predict the probability of failure. It is found that the system probability of failure is influenced more by delamination than the failure due to matrix cracking. Shear strength (T12) and Young’s modulus (E1 and E3) are the most sensitive parameters to influence the composite plate reliability.

- by Shivdayal Patel and +1
- •
- Structural Reliability, Stochastic finite elements
- by Drahomir Novak
- •
- Engineering, Mechanical Engineering, Civil Engineering, Engineering Mechanics

The intrusive polynomial chaos (PC) approach for uncertainty quantification in numerous engineering problems constitutes a computationally challenging task. Indeed, the spectral stochastic finite element method (SSFEM) leads to a large-scale deterministic linear system for the PC coefficients of the solution process. When the underlying physical problem is already large, domain decomposition techniques are a natural way to split the problem into a set of smaller subproblems and solve them concurrently on multiprocessor computers. FETI-DP and BDDC domain decomposition techniques for the SSFEM have recently proposed in Subber and Sarkar [2013, 2014]. In this paper, we
formulate overlapping domain decomposition for the solution of the large-scale linear system in the SSFEM. In the Schwarz preconditioner, the global vertices of the physical domain are split into (preferably, but not necessarily overlapping) subsets which constitute the local subdomains. Based on these subsets, restriction operators are defined to extract the local PC coefficients of the solution process from the global one. The restriction operators associated with the local vertices are then used to construct the local stochastic stiffness matrix of each subdomain as block of the global stiffness matrix. Consequently, stochastic Dirichlet problems corresponding to the local stiffness matrices can be solved on each subdomain concurrently. The solution of these local Dirichlet problems are used to defined the stochastic Schwarz preconditioner. It turns out that the one-level stochastic Schwarz preconditioner can be viewed as a parallel generalization of the block-diagonal mean based preconditioner Powell and Elman [2009], whereby the associated deterministic problems are solved in parallel using the deterministic Schwarz
preconditioner. A coarse grid correction providing a mechanism to propagate information across the subdomains globally is supplied to the one-level Schwarz preconditioner. This global exchange of information leads to a scalable performance for large number of subdomains. For the numerical illustrations, a two dimensional elliptic SPDE with spatially varying random coefficients is considered. Numerical scalability of the algorithm is investigated
with respect to dimension and order of the stochastic expansion, strength of the input uncertainty and the geometric parameters.

- by Ehsan Adeli
- •
- Mechanical Engineering, Civil Engineering, Computer Science, Computational Mechanics
- by Jeroen Witteveen
- •
- Engineering, Monte Carlo Simulation, Time Series, Modeling
- by Jungwon Huh
- •
- Engineering, Monte Carlo Simulation, Stochastic finite elements, Reliability Analysis

This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation.

A state of the art on simulation methods in stochastic structural analysis is presented. The purpose of the paper is to review some of the different methods available for analysing the effects of randomness of models and data in structural analysis. While most of these techniques can be grouped under the general name ofMonte Carlo methods, the several published algorithms are more suitable to some objectives of analysis than to others in each case. These objectives have been classified into the foolowing cathegories: (1), TheStatistical Description of the structural scattering, a primary analysis in which the uncertain parameters are treated as random variables; (2) The consideration of the spatial variability of the random parameters, that must then be modelled as Random Fields (Stochastic Finite Elements); (3) The advanced Monte Carlo methods for calculating the usually very low failure probabilities (Reliability Analysis), and, (4), a deterministic technique that depart from the random nature of the above methods, but which can be linked with them in some cases, known as theResponse Surface Method. All of these techniques are critically examined and discussed. The concluding remarks point out some research needs in the field from the authors' point of view.

One source of microstructural variability in fiber reinforced polymers is variability in fiber volume fraction. Despite large scatter in fiber volume fraction, an average value is usually reported without any consideration regarding the variability. Significant variation in fiber volume fraction across the sample reveals that homogeneity throughout the sample is a poor assumption and the commonly employed representative volume element is a poor representation of real microstructures. In this work, the distributions of fiber volume fraction at different length scales are investigated. The variation in distributions suggests that a length scale-dependent distribution must be selected in probabilistic analysis. A cross-correlation between fiber volume fractions of nearest neighbors was computed and an uncorrelated volume element (UVE) is introduced for the length scale at which fiber volume fractions become uncorrelated. The concept of a UVE enables random attribution of properties in stochastic modeling.

- by Ray Fertig and +1
- •
- Composite Materials and Structures, Digital Image Processing, Failure Analysis, Polymer Composites
- by Martin Ostoja-Starzewski
- •
- Stochastic finite elements
- by Ehsan Adeli
- •
- Mechanical Engineering, Probability Theory, Computer Science, Computational Mechanics
- by Abdallah Mostafa
- •
- Applied Mathematics, Finite Element, Stochastic finite elements, Case Study

A generalized Bernoulli method is used for constructing new exact soliton solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Bernoulli equation which has a simple exponential solution. Five important models in mathematical physics named, the nonlinear dispersive equation, the nonlinear Fisher-type equation, ZK-BBM equation, the general Burgers-Fisher equation and Drinfeld–Sokolov system are investigated. We successfully get new soliton solutions for these problems and recover some solutions that had been found by other methods for the same problems.

- by Abdallah Mostafa and +1
- •
- Stochastic finite elements, Nonlinear Partial Differential Equations, Stochastic differential equation
- by Nikos Lagaros and +1
- •
- Engineering, Monte Carlo Simulation, Mathematical Sciences, Stochastic finite elements

The design of earth structures requires the calculation of soil deformation which is generally a difficult task because of the uncertainty and spatial variability of the properties of soil materials. This paper presents a procedure of conducting Stochastic Finite Element Analysis using Polynomial Chaos in order to quantitative the uncertainties. Among other methods the procedure leads to an efficient computational cost for real practical problems. This is achieved by polynomial chaos expansion of the displacement field. An example of a plane-strain strip load on a semi-infinite elastic foundation is presented and the results of settlement are compared to those obtained from the closed form solution method. A close matching of the two is observed.

We present a stochastic finite-element approach for characterizing parameter dependence of minimum eigenvalue problems encountered in neutronic calculations. Our formulation results in solving a nonlinear system of equations, that is K times larger than the original problem and has K constraints, where K is the number of terms considered in the perturbative expansion of the solution. This approach allows us

- by Monika Neda
- •
- Mathematics, Monte Carlo, Sensitivity Analysis, Stochastic finite elements
- by miguel gutierrez
- •
- Engineering, Stochastic finite elements
- by Sankaran Mahadevan
- •
- Engineering, Finite Element, Mathematical Sciences, Stochastic finite elements
- by F. Schoefs and +1
- •
- Engineering, Numerical Simulation, Mathematical Sciences, Stochastic finite elements

ABSTRACT In the design of micro-electromechanical systems (MEMS) such as micro-resonators, one of the major dissipation phenomena to consider is thermoelastic damping. The performance of such MEMS is directly related to their thermoelastic quality factor which has to be predicted accurately. Moreover, the performance of MEMS depends on manufacturing processes which cause substantial uncertainty in the geometry and in the material properties of the device. The aim of this paper is to provide a framework to account for uncertainties in the finite element analysis of thermoelastic damping in MEMS. Four stochastic methods are investigated: perturbation Stochastic Finite Element Method (SFEM), spectral SFEM, projection method and Monte-Carlo method. Due to the nature of the thermoelastic problem, these stochastic methods are extended to the resolution of non-symmetric damped eigenproblems. The methods are applied on the analysis of the thermoelastic quality factor of a micro-beam whose elastic modulus is considered as random.

- by Kurt Maute
- •
- Stochastic finite elements, Quality Factor, Elastic Modulus, Projection Method

The technique of stochastic finite element (SFEM) which is the finite element technique FEM adapted to stochastic problems can be re-described to use random variable transformation technique RVT. A new FEM-RVT technique was successfully used in solving stochastic problems with random excitation [M. El-Tawil, W. El-Tahhan, A. Hussein, A proposed technique of SFEM on solving ordinary random differential equation, J. Appl. Math. Comput. 161 (2005) 35-47]. In this paper, the technique is adapted to solve a randomly excited differential equation with a random operator. The technique shows high accuracy when solving a case study compared with the exact solution. Finally a problem with unknown exact solution is solved using this technique.

- by Wahba El-Tahan
- •
- Mathematics, Applied Mathematics, Computer Science, Finite Element
- by Abdallah Zaki
- •
- Mathematics, Applied Mathematics, Computer Science, Finite Element
- by Abdallah Mostafa
- •
- Applied Mathematics, Finite Element, Stochastic finite elements, Case Study
- by Nikos Lagaros
- •
- Civil Engineering, Monte Carlo Simulation, Optimization Problem, Structural Safety

Stochastic performance measures can be taken into account, in structural optimization, using two distinct formulations: robust design optimization (RDO) and reliability-based design optimization (RBDO). According to a RDO formulation, it is desired to obtain solutions insensitive to the uncontrollable parameter variation. In the present study, the solution of a structural robust design problem formulated as a two-objective optimization problem is addressed, where cross-sectional dimensions, material properties and earthquake loading are considered as random variables. Additionally, a two-objective deterministic-based optimization (DBO) problem is also considered. In particular, the DBO and RDO formulations are employed for assessing the Greek national seismic design code for steel structural buildings with respect to the behavioral factor considered. The limit-state-dependent cost is used as a measure of assessment. The stochastic finite element problem is solved using the Monte Carlo Simulation method, while a modified NSGA-II algorithm is employed for solving the two-objective optimization problem.

- by Nikos Lagaros
- •
- Engineering, Monte Carlo Simulation, Steel Structure, Seismic Design
- by Mohammad Tawfik and +1
- •
- Mechanical Engineering, Civil Engineering, Computational Mechanics, Monte Carlo Simulation

This paper deals with several numerical techniques that account for random excitations and random material parameters occurring in earthquake engineering. It focuses in sequence on the influence on the structural response of the variability of the incident field using filtering theory, on the soil variability by coupling Stochastic Finite Elements, integral operators and Monte-Carlo simulation, and finally on the influence

- by Didier Clouteau and +1
- •
- Applied Mathematics, Earthquake Engineering, Monte Carlo Simulation, Seismic Engineering
- by Alex H. Barbat
- •
- Engineering, Computer Science, Stochastic Mechanics, Mathematical Sciences
- by Bullet Baroth
- •
- Engineering, Numerical Method, Mathematical Sciences, Stochastic finite elements