A spectral stochastic element free Galerkin method for the problems with random material parameter

This paper presents a spectral stochastic element free Galerkin method (SSEFGM) for the problems involving a random material property. The random material property and resulting system response quantity are represented by a probabilistic spectral expansion techniques (Karhunen–Loeve expansion and Polynomical Chaos series, respectively) and implemented into the element free Galerkin (EFG) analysis. Numerical solutions in 1D linear elastic problem with random elastic modulus are introduced, and compared with those of Monte Carlo simulation (MCS) so as to provide the validation of the proposed approach. The present SSEFGM approach can produce a probabilistic density distribution as well as a first‐ and second‐order statistical moments (mean and variance) of response quantity by a single calculation, which is distinguished from an iterative MCS. Moreover, the method is based on an element free analysis so that there is no need of nodal connectivities, which usually require more time and labourious task than main calculations. Thus the proposed SSEFGM approach can provide an alternative analysis tool for the problems contains a stochastic material property, and demands complex mesh structures. Copyright © 2004 John Wiley & Sons, Ltd.     

On the efficacy of stochastic collocation,stochastic Galerkin,and stochastic reduced order models for solving stochastic problems

The stochastic collocation (SC) and stochastic Galerkin (SG) methods are two well-established and successful approaches for solving general stochastic problems. A recently developed method based on stochastic reduced order models (SROMs) can also be used. Herein we provide a comparison of the three methods for some numerical examples; our evaluation only holds for the examples considered in the paper. The purpose of the comparisons is not to criticize the SC or SG methods, which have proven very useful for a broad range of applications, nor is it to provide overall ratings of these methods as compared to the SROM method. Rather, our objectives are to present the SROM method as an alternative approach to solving stochastic problems and provide information on the computational effort required by the implementation of each method, while simultaneously assessing their performance for a collection of specific problems.     

Coupling techniques in boundary-combined methods for solving elliptic problems with singularities

Three coupling strategies in matching the Ritz-Galerkin method and the finite element method are introduced for general elliptic equations, and useful numerical techniques are provided. Numerical experiments have been carried out for solving the typical, singular Motz problem, which shows that optimal convergence rates of numerical solutions can be achieved by using the combined methods and techniques provided in this paper.     

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.     

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the Chapman–Kolmogorov (C–K) equation. This is accomplished by circumventing the solution of the associated Euler–Lagrange equation ordinarily used in the path integral based procedures. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.     

Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions

In this paper, the two-dimensional Legendre wavelets are applied for numerical solution of the fractional Poisson equation with Dirichlet boundary conditions. In this way, a new operational matrix of fractional derivative for the Legendre wavelets is derived and then this operational matrix has been employed to obtain the numerical solution of the above-mentioned problem. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence of the two-dimensional Legendre wavelets expansion is investigated. Also the power of this manageable method is illustrated.     

Stochastic FEM sensitivity analysis of nonlinear dynamic problems

This paper deals with the stochastic finite element analysis of nonlinear structural dynamic problems, consisting of a linearly elastic square plate lying on a nonlinear foundation and loaded with a deterministic uniform transverse dynamic load. The plate can be either simply-supported or fixed-all-around. The stochasticity of the problem arises from the spatial randomness of the elastic modulus of the plate and/or from the spatial randomness of a coefficient controlling the degree of nonlinearity of the foundation. Monte Carlo simulation techniques along with a finite element formulation of the problem are used in order to analyze the system. It is concluded that the maximum deflection at the center of the square plate fits to the lognormal distribution. Various conclusions are also drawn on the influence of the stochasticity of the elastic modulus and/or the stochasticity of the nonlinear foundation and the degree of nonlinearity of the foundation on the value of the coefficient of variation of the maximum deflection at the center of the plate.     

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.     

A Monte Carlo simulation based heuristic procedure for solving dynamic line layout problems for facilities using conventional material handling devices

A simple procedure is proposed to identify line layout solutions when a production facility with work centres of unequal size uses conventional material handling devices and operates under stochastic demand scenarios. The procedure uses Monte Carlo simulation (MCS) to empirically search for robust solutions defined as those that simultaneously meet minimum material handling cost performance levels across all demand scenarios. The results reported in this study suggest that ‘robust’ line layout solutions can be identified using a modest volume of random sampling. The procedure and results are demonstrated through a series of sample problems.     

A symmetric Galerkin BEM implementation for 3D elastostatic problems with an extension to curved elements

 Like the finite element method (FEM), the symmetric Galerkin boundary element method (SGBEM) can produce symmetric system matrices. While widely developed for two dimensional problems, the 3D-applications of the SGBEM are very rare. This paper deals with the regularization of the singular integrals in the case of 3D elastostatic problems. It is shown that the integration formulas can be extended to curved elements. In contrast to other techniques, the Kelvin fundamental solutions are used with no need to introduce the new kernel functions. The accuracy of the developed integration formulas is verified on a problem with known analytical solution. Received 6 November 2000     

Optimization of aluminium sheet hot stamping process using a multi-objective stochastic approach

This article aims to investigate the means to obtain optimal hot stamping process parameters and the influence of the stochastic variability of these parameters on forming quality. A multi-objective stochastic approach, integrating response surface methodology (RSM), multi-objective genetic algorithm optimization non-dominated sorting genetic algorithm II (NSGA-II) and the Monte Carlo simulation (MCS) method is proposed in this article to achieve this goal. RSM was used to establish the relationship between the process parameters and forming quality indices. NSGA-II was utilized to obtain a Pareto frontier, which consists of a series of optimal process parameters. The MCS method was employed to study and reduce the influence of a stochastic property of these process parameters on forming quality. The results confirmed the efficiency of the proposed multi-objective stochastic approach during optimization of the hot stamping process. Robust optimal process parameters guaranteeing good forming quality were also obtained using this approach.     

On the effectiveness of Monte Carlo simulation and heuristic search for solving large-scale block layout problems

The effectiveness of Monte Carlo simulation relative to intelligent search strategies for solving block layout problems is investigated. For testing purposes, 810 block layout problems are constructed to span a wide range of problem sizes, material flow variation levels, work centre space requirements distributions, and work centre shape distributions. Contrary to preliminary results reported in earlier studies, greedy search and simulated annealing consistently outperform Monte Carlo Simulation across the full range of test problems and sample sizes. This divergence is explained through a comparison based on probabilistic derivations between the proportion of good solutions sampled by the Monte Carlo method and those found by the heuristic search methods. Conditions for the superiority of either method are identified. Therefore, the current study complements earlier studies by providing analytical arguments and additional experimental evidence for the effectiveness of simple Monte Carlo method and intelligent search heuristics on solving layout problems.     

An interface integral equation method for solving general multi‐medium mechanics problems

In this paper, based on the general stress–strain relationship, displacement and stress boundary‐domain integral equations are established for single medium with varying material properties. From the established integral equations, single interface integral equations are derived for solving general multi‐medium mechanics problems by making use of the variation feature of the material properties. The displacement and stress interface integral equations derived in this paper can be applied to solve non‐homogeneous, anisotropic, and non‐linear multi‐medium problems in a unified way. By imposing some assumptions on the derived integral equations, detailed expressions for some specific mechanics problems are deduced, and a few numerical examples are given to demonstrate the correctness and robustness of the derived displacement and stress interface integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical stability in regular BEM schemes for elastostatic problems

The regular boundary element method is employed for the static analysis of boundary value problems of elasticity. This method allows one to reduce a given boundary value problem to a system of regular integral equations of the first kind with respect to source functions not located on the boundary. This paper is concerned with the numerical stability analysis of regular boundary element methods. In particular, the existence and stability of approximate solutions for integral equations of the first kind with continuous kernels are discussed. The special regularization technique for treating such a class of integral equations is developed. Numerical examples illustrate proposed algorithms and demonstrate their advantages.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Scalar wave equation by the boundary element method: a D-BEM approach with non-homogeneous initial conditions

This work is concerned with the development of a D-BEM approach to the solution of 2D scalar wave propagation problems. The time-marching process can be accomplished with the use of the Houbolt method, as usual, or with the use of the Newmark method. Special attention was devoted to the development of a procedure that allows for the computation of the initial conditions contributions. In order to verify the applicability of the Newmark method and also the correctness of the expressions concerned with the computation of the initial conditions contributions, four examples are presented and the D-BEM results are compared with the corresponding analytical solutions.     

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

For an isotropic linear elastic body, only displacement or traction boundary conditions are given on a part of its boundary, whilst all of displacement and traction vectors are unknown on the rest of the boundary. The inverse problem is different from the Cauchy problems. All the unknown boundary conditions on the whole boundary must be determined with some interior points' information. The preconditioned conjugate gradient method (PCGM) in combination with the boundary element method (BEM) is developed for reconstructing the boundary conditions, and the PCGM is compared with the conjugate gradient method (CGM). Morozov's discrepancy principle is employed to select the iteration step. The analytical integral algorithm is proposed to treat the nearly singular integrals when the interior points are very close to the boundary. The numerical solutions of the boundary conditions are not sensitive to the locations of the interior points if these points are distributed along the entire boundary of the considered domain. The numerical results confirm that the PCGM and CGM produce convergent and stable numerical solutions with respect to increasing the number of interior points and decreasing the amount of noise added into the input data.     

An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods

Conventional numerical implementation of the boundary element method (BEM) for elasto-plastic analysis requires a domain discretization into cells. This requires more effort for the discretization of the problem and additional computational effort. A new technique is proposed here for the analysis of 2D and 3D elasto-plastic problems with the boundary element method. In this approach the domain does not need to be discretised into cells prior to the analysis. Plasticity is assumed to start from the boundary and the cells are generated from the boundary data automatically during the analysis. Using the cell generation process, elasto-plastic analysis with the BEM becomes much more user friendly and efficient than the standard approach with a pre-definition of cells. The accuracy and efficiency of the solution obtained by the new approach is verified by several numerical examples.     

Modeling of viscoelastic structures with random material properties using time-separated stochastic mechanics

Modeling and simulation of materials with stochastic properties is an emerging field in both mathematics and mechanics. The most important goal is to compute the stochastic characteristics of the random stress, such as the expectation value and the standard deviation. An accurate approach are Monte Carlo simulations; however, they consume drastic computational power due to the large number of stochastic realizations that have to be simulated before convergence is achieved. In this paper, we show that a recently published approach for accurate modeling of viscoelastic materials with stochastic material properties at the material point level in the work of Junker and Nagel is also valid for macroscopic bodies. The method is based on a separation of random but time-invariant variables and time-dependent but deterministic variables for the strain response at the material point (time-separated stochastic mechanics [TSM]). We recall the governing equations, derive a simplified form, and discuss the numerical implementation into a finite element routine. To validate our approach, we compare the TSM simulations with Monte Carlo simulations, which provide the “true” answer but at unaffordable computational costs. In contrast, the numerical effort of our approach is in the same range as for deterministic viscoelastic simulations.     

On the dual iterative stochastic perturbation‐based finite element method in solid mechanics with Gaussian uncertainties

The main idea is a dual mathematical formulation and computational implementation of the iterative stochastic perturbation‐based finite element method for both linear and nonlinear problems in solid mechanics. A general‐order Taylor expansion with random coefficients serves here for the iterative determination of the basic probabilistic characteristics, where linearization procedure widely applicable in stochastic perturbation technique is replaced with the iterative one. The expected values and, in turn, the variances are derived first, and then, they are substituted into the equations for higher central probabilistic moments and additional probabilistic characteristics. The additional formulas for up to the fourth‐order probabilistic characteristics are derived thanks to the 10th‐order Taylor expansion. Computational implementation of this idea in the stochastic finite element method is provided by using the direct differentiation method and, independently, the response function method with polynomial basis. Numerical experiments include the simple tension of the elastic bar, nonlinear elasto‐plastic analysis of the aluminum 2D truss, and solution to the homogenization problem of periodic fiber‐reinforced composite with random elastic properties. The expected values, coefficients of variation, skewness, and kurtosis of the structural response determined via this iterative scheme are contrasted with these estimated by the Monte Carlo simulation as well as with the results of the semi‐analytical probabilistic technique following the response function method itself. Although the entire methodology is illustrated here by using the Gaussian variables where all odd‐order terms simply vanish, it can be extended towards non‐Gaussian processes as well and completed with all the perturbation orders. Copyright © 2015 John Wiley & Sons, Ltd.